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Cycle structures of permutations with restricted positions
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Cycle structures of permutations with restricted positions
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Cycle Structures of Permutations with Restricted Positions by Enes Ozel A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (Applied Mathematics) May 2018 Copyright 2018 Enes Ozel Acknowledgements I would like to thank all my professors and colleagues at the University of Southern California, Department of Mathematics, for their continuous support throughout my doctoral education. I am indebted to my advisor, Jason Fulman, for his endless support; he provided me with interesting problems to tackle, pushed me in the right direction when I was stuck, and always gave reliable guidance. I cannot overemphasize the contributions of my professors to my academic formation. I offer my gratitude to Professors Jason Fulman, Larry Goldstein and Richard Arratia for the strong mathematical intuition and extensive theoretical knowledge they provided in numerous courses. I would also like to thank Professors Fulman, Arratia and Paul Newton for their presence in my dissertation committee. I have cherished the comforting and supporting company of my office mates and close friends Can Ozan Oğuz, Ezgi Kantarcı Oğuz, İlknur Eğilmez, Hilmi Eğilmez, Emre Demirkaya, Duygu Kaba, Melike Şırlancı, Onur Tüysüzoğlu, Güher Çamlıyurt, Ümit Baş and Ahsan Javed, through these six years. I know that wherever they may end up, they will become most proficient persons of letters. I would like to extend my gratitude to The Scientific and Technological Research Council of Turkey, TÜBİTAK, for financially supporting me between 2009 - 2011 and again between 2011 - 2012, during the early years of my graduate education. I would like to thank my father, Ahmet Özel, whose dedication and conscientiousness in his scholarly work I have always admired and emulated. I am eternally grateful to my sisters, Betül Özel Çiçek and Esma Özel, who were, as they have always been, great sources of intellectual and emotional wisdom. ii Lastly, I would like to thank my wife and best friend, Hacer Şifanur Özel. She constantly strives to bring more beauty and peace to our lives. She was my greatest source of comfort and support during the worst and best of times. She is the tear that hangs inside my soul forever. This dissertation is dedicated to the memory of my mother, Ayşe Bengigül Özel, who passed away in April 2016. Without her encouragement and moral support I would not even have thought to undertake this journey of becoming a professional learner and teacher. iii Table of Contents Acknowledgements ii List Of Tables vi List Of Figures vii Abstract viii Chapter 1: Introduction 1 1.1 Overview of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Chapter 2: Permutations with Restricted Positions 6 2.1 Definitions and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.1 Cycle Index for S n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.2 Expected Value and Asymptotic Distribution of Number of k-Cycles . . . . . . . 11 2.2 Restriction Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.1 Permanent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Two-sided Restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.4 One-sided Restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4.1 Moment Calculations for Fixed Points . . . . . . . . . . . . . . . . . . . . . . . . 25 Chapter 3: b (r) -Regular Permutations and Compositions 29 3.1 b (r) -Regular Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.1.1 b (2) -Regular Permutations and k-Cycles . . . . . . . . . . . . . . . . . . . . . . . 32 3.2 Compositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.2.1 The Bijection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.2.2 Some Results Regarding Parts of Compositions . . . . . . . . . . . . . . . . . . . 37 3.3 Cycle Index of b (2) -Regular Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Chapter 4: k-Cycle Indicators, m-Dependence and the Main Result 47 4.1 k-cycle Indicators of b (2) -regular Permutations . . . . . . . . . . . . . . . . . . . . . . . . 48 4.1.1 Dependence Structure of the Indicators . . . . . . . . . . . . . . . . . . . . . . . 49 4.1.2 m-dependence and Central Limit Theorems . . . . . . . . . . . . . . . . . . . . . 52 4.2 The Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.2.1 Probability Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.2.2 Two Graph Dependency CLTs and the Result . . . . . . . . . . . . . . . . . . . . 56 Chapter 5: Conclusion and Future Work 62 5.1 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.2 Colored Compositions and the Random Walk Structures . . . . . . . . . . . . . . . . . . 66 5.2.1 Random Transposition Walk on Restricted Permutations . . . . . . . . . . . . . 69 5.3 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 iv Reference List 76 Appendix A Matlab Code for Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 v List Of Tables 5.1 b (3) -regular permutations off1; 2; 3; 4g . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.2 Two-colored compositions of 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 vi List Of Figures 4.1 Dependency Graph for Fixed Point Indicators . . . . . . . . . . . . . . . . . . . . . . . . 51 4.2 Dependency Graph for Transposition Indicators . . . . . . . . . . . . . . . . . . . . . . . 51 5.1 Histogram of fixed points for b (3) -regular permutations on 1000 letters . . . . . . . . . . 65 5.2 Histogram of fixed points for b (5) -regular permutations on 1000 letters . . . . . . . . . . 65 5.3 Histogram of fixed points for b (100) -regular permutations on 10000 letters . . . . . . . . 66 vii Abstract In this dissertation we study cycles of permutations with restricted positions. Permutations with restricted positions have strong connections to many other fields of mathematics, including Complexity Theory and Graph Theory, as well as applications in Coding Theory and Mathematical Statistics. Our objective is to identify the relationship between the positional restrictions and the cycle structures. Specifically, we aim to identify the asymptotic distribution of the random variable C n;k , the number of k-cycles of a uniformly randomly selected permutation adhering to the given positional restriction. For the general one-sided restrictions we provide a method to calculate the moments of the C n;1 , the number of fixed points. Considering a certain family of one-sided restrictions we prove the central limit theorems for C n;k , establishing the Normal asymptotic distribution with corresponding values of mean and variance. To this end, we utilize an application of Stein’s Method for locally dependent random variables. Further, our investigation leads to the discovery of bijections that relates permutations with restricted positions to compositions of positive integers and their generalizations. These connections allow the translation of results in one set of objects to the other, providing another important motivation to further this inquiry. viii Chapter 1 Introduction There are various ways to answer the question “What do given combinatorial objects look like?”. The first natural question to ask is how many such objects there are. An explicit formula or a generating function would be ideal; if not a recurrence relation might be a decent substitute. Next, one might seek after specific properties of these objects, such as the nature of certain statistics or invariants. It might be that we are interested in the number of these objects showing some peculiar behavior. In this dissertation the objects our interest are permutations with restricted positions. Despite being relatively easy to visualize with the help of 0 1 matrices, these objects show fundamental but counter intuitive connections to many other fields of mathematical inquiry. Many classical problems in probability theory and combinatorics may be modeled using these positional restrictions, suggesting new approaches and solutions. The problem of counting permutations with restricted positions for a general restrictional description turns out to be an important problem for Theory of Computation (see [35]). Further, there are structural connections to Rook Theory, Matching Theory, Random Walks and Markov Chains, Lattices, Compositions and Lattices, some of which we will encounter throughout this work. To top it all off, there are concrete applications of permutations with restricted positions in Coding Theory (see [28], [6]), as well as tests of independence under truncated data (see [18]). A great survey of some of these connections could be found in the 2001 paper “Statistical Problems Involving Permutations with Restricted Positions” by Diaconis, Graham and Holmes [13]. 1 Despite these connections and applications, the study of these objects remains an untapped field. In this work we would like to initiate the extensive program of understanding these objects fully. There are, again, many ways to study the structure of a permutation. Analysis of certain statistics, such as number of inversions, number of cycles, number of rises and falls, number of specific patterns that occur has an extensive literature. We will pursue the problem of counting the number of cycles of a given size. Whatever restriction we consider, we will select a corresponding permutation uniformly at random and investigate the number of k-cycles. For certain families of restrictions we calculate different moments of these random variables and establish the asymptotic distribution. Further, during this investigation, we discover strong connections to other combinatorial objects, such as compositions, and the fact that many results related to permutations with restricted positions translate to these objects. This fact provides a motivation to identify further connections with the objective of establishing results in one set of objects using the structure in the other set of objects. In the following section we provide a brief summary of the dissertation and our contributions to the aforementioned problems. 1.1 Overview of the Dissertation We begin by introducing our main definitions and notations regarding permutations in Chapter 2. We then define and derive the cycle index for the whole symmetric group, and show some applications of the cycle index as a technique we also use in permutations with restricted positions. Next, we move on to a general description of permutations with restricted positions and 0 1 restriction matrices. The discussion naturally leads to the problem of calculating the permanent of a 0 1 matrix and to the Theory of Computation. This section is a brief indicator of the fact that problems related to restrictions might be intrinsically difficult. The following section is on two-sided restrictions, where the range for each domain element of a permutation is bounded both above and below. As a working example we study the Fibonacci permutations and answer the same question we ask throughout the dissertation: “What is the asymptotic distribution of the number of k-cycles for permutations adhering 2 to this type of restriction?” Then we proceed to one-sided restrictions and provide a detailed analysis of these permutations. In particular, we establish a new method to calculate the moments of the random variable, number ofk-cycles of a uniformly randomly selected one-sided restricted permutation. Using this method, we calculate the expected value and variance for the number of fixed points for any one-sided restricted permutation. This is Theorem 2.4.6 in Chapter 2. Theorem. Let be a randomly chosen permutation with (i) b i for some given b i 2f1; 2;:::;ng. Then E[C n;1 ()] = 1 jS b j n X k=1 jS b;k j and Var(C n;1 ()) = 1 jS b j n X k=1 h S b;k (S b S b;k ) i + 2 X i<j h S b;i;j S b;i S b;j i ; wherejS b n j = Q n i=1 (1 + (ib i )),jS b;k j = k1 Q j=1 (1 +jb j ) n Q j=k+1 (j (b j 1(b j >k))) and S b;i;j = i1 Y m=1 1 +mb m j1 Y m=i+1 1 +m (b m+1 1(b m+1 >i)) n Y m=j+1 1 +m (b m+2 1(b m+2 >i)) 1(b m+2 >j)) : This, and many other results we establish are presented in our 2017 preprint paper, titled “The Number of k-Cycles In a Family of Restricted Permutations” [30]. Chapter 3 focuses on the main family of restrictions of our interest, one-sided b (r) restrictions. Here we would like (j) 8 > > > < > > > : 1; if jr jr + 1 if j >r; specifically when r = 2. We identify the nature of the potential cycles for these permutations and proceed to a section regarding compositions of integers. We construct an interesting bijection between 3 the set of permutations with b (2) restriction, or b (2) -regular permutations as we will call them, and the compositions of n. This bijection has the property that k-cycles are mapped to k-parts alongside permutations to compositions. This property enables us to translate any result we prove on cycles of permutations to parts of compositions. We then present some of the results present in the composition literature that can be translated back accordingly to results about permutations. In Section 3.3, we construct the cycle index for b (2) -regular permutations using this bijection and calculate the first mo- ments of the number of k-cycles for a uniformly randomly selected b (2) -regular permutation. This is a crucial step in establishing the asymptotic distributional results for these permutations. The following is Theorem 3.3.4 in Chapter 3. Theorem. LetC n;k be the number ofk-cycles of a uniformly randomly selectedb (2) -regular permutation. Then the expected value for C n;k is E[C n;k ] = nk + 3 2 k+1 ; and the variance is Var(C n;k ) = (2 k+1 2k + 3)n + 3k(k 4) + (3k)2 k+1 + 5 4 k+1 : Chapter 4 is the partwhere we take the finalstepsof proving our main asymptotic result, namely, the central limit theorem (CLT) that statesC n;k has the Normal limiting distribution with the above mean and variance. We start this chapter by identifying the dependence structure of the k-cycle indicator variables, discovering a curious local dependency structure. It turns out that this local dependency structure is a special case of m-dependence, for which literature is already vast. Then, we present a brief overview of these existing results. We then turn our attention to the probability metrics, with which we set up the upper bounds to the distributional distance between the target Normal distribution and our shifted and scaled version of C n;k . 4 Provingm-dependence gets exponentially more complicated. This forces us to reconsider the initial form of local dependence. This form of dependence might be expressed using a dependency graph, where the nodes are the random variables and there is an edge between two nodes if the random variables are dependent. We use an application of the infamous Stein’s Method that utilizes such form of dependence. There are two such results we compare, and it turns out that Chen and Shao’s [10] local dependence CLT provides much stronger upper bounds. Thus, our main result makes use of this result and is stated as follows (Theorem 4.2.5). Theorem. Let be a uniformly randomly selected b 2 -regular permutation, Y k i be the shifted and scaled indicators of k-cycles of and 2 = 2 n;k . Further, define W = 1 P i Y k i . Then for ZN(0; 1), d K (W;Z) 75 (2k) 10 (nk + 1) 1 1=2 k 3 : We can reorganize this upper bound as d K (W;Z)f(k) nk + 1 3 n;k ; where f(k) = 75 (2k) 10 (1 1=2 k ) 3 . As 3 n;k = (n 3=2 ), the upper bound is essentially of order (n 1=2 ). The last chapter, Chapter 5 contains our investigation and insights as to why our main result can be generalized for b (r) -regular permutations, with r being any fixed positive integer greater than 2. These investigations provides connections to colored compositions and the random transposition walks on permutations, leading to open research problems about these random walks. 5 Chapter 2 Permutations with Restricted Positions In this chapter we will introduce the main objects of interest for this dissertation, permutations with restricted positions. We will start by presenting the most general way of defining positional restrictions, i.e., through 01 matrices. The discussion of 01 matrices naturally leads to discussion of permanents, so that will be our next stop. Calculation of the permanent of a 0 1 matrix is a difficult problem, and we will expound this in some detail. After that we will look to specific types of restrictions, namely, two-sided and one-sided restrictions, providing the motivation and examples related to these forms of restrictions. For one-sided restrictions we will develop a method based on permanents to calculate the moments for number of fixed points of a randomly selected restricted permutation. Then, as a special case of one-sided restrictions we introduce b (r) -regular permutations and present some initial overview of these permutations, eventually for which we aim to understand the cycle behavior. We finish the chapter by providing an application of our permanent method on these b (r) -regular permutations and calculate the expected value and variance of their number of fixed points. 2.1 Definitions and Notation There are various technical and intuitive ways to define and analyze the symmetric group in algebra and geometry, but we will take the following approach. 6 Definition 2.1.1 (Permutation). Let [n] = f1; 2;:::;ng and define S n to be the set of bijections : [n] ! [n]. S n is called the symmetric group on n numbers and the elements of S n are called permutations. For a given permutation , assume we have (i) = j. Here, i will be referred to as “position” or “coordinate”, whereas j will be called “label”. So, when we say “Label 5 is allowed at position 4.”, we mean “(4) = 5” is potentially possible. For the next definition we need a shorthand notation for repeatedly composing with itself, i.e., for any positive integer k, k (x) = ((:::(x):::)), where are exactly k many ’s composed with each other. For example, 2 (i) =((i)). Definition 2.1.2 (k-cycle). Let 2 S n , i2 [n]. Then if there exists a positive integer k such that k (i) =i, then we call (i (i) 2 (i) :::; k1 (i)) a k-cycle of . This dissertation focuses on the number of k-cycles of permutations that satisfy specific conditions. We will adopt two basic ways to express a permutation , using both under different contexts. The first and most basic style is Cauchy’s two-line notation, where mapping of elements of [n] to [n] is shown by listing the list in two lines, the above line corresponding to positions and below line corresponding to labels. Example 2.1.3. Let 2 S 5 such that (i) = i + 1(mod 5). In the two-line notation this would be expressed as = 1 2 3 4 5 2 3 4 5 1 : When all the cycles, written as in Definition 2.1.2, of a permutation are listed consecutively, this is referred to as cycle notation. If the objective is to study the cycles of permutations, the second notation style is usually preferred. Example 2.1.4. Let be as in Example 2.1.3. Then the cycle notation for would be = (1 2 3 4 5): 7 Here one can imagine the notation as “1 goes to 2”, “2 goes to 3”, so on and so forth. Example 2.1.5. Let = 1 2 3 4 5 3 5 1 2 4 . Then the cycle notation would be = (1 3)(2 5 4): It is customary to list the cycles according to the minimum label they include. 2.1.1 Cycle Index for S n Now that we have a way to express cycles of permutations efficiently, we turn our attention to cycle index of S n . Let c k :=c k () denote the number of k-cycles for a permutation . Then it is clear that 1c 1 + 2c 2 +::: +nc n =n: Following Stanley’s (2011) [34] suit, we will call a permutation to be of type (or, cycle type) C = (c 1 ;c 2 ;:::;c n ), if has c k many k-cycles for each k2f1; 2;:::;ng. Note that such a vector C naturally induces an partition of n. A partition of n is an unordered list of positive integers that add up to n. Example 2.1.6. Consider the permutation expressed in cycle notation, = (165)(23)(4)(7). Then 2S 7 is of typeC = (2; 1; 1). In this case the corresponding partition of 7 would be 3 + 2 + 1 + 1. A natural question to be asked at this stage is, for a given cycle type C, how many permutations 2S n are there of this type? Proposition 2.1.7 (Proposition 1.3.2 in Stanley (2011)). There are n!=1 c1 c 1 ! 2 c2 c 2 !:::n cn c n ! many permutations of type C = (c 1 ;c 2 ;:::;c n ). Proof. Let 2S n be any permutation of type C. By permuting the positions of labels of we would getn! permutations of typeC, some of which are repeated. Now, there are two ways a repetition might occur. 8 (i) If the same set of labels fall within the same cycle, then repetition occurs when two different ar- rangements correspond to the same specific cycle with different starting elements, e.g., (132) versus (321). For each k-cycle this causes k many repetitions. This results in a total of 1 c1 2 c2 :::n cn many repetitions. (ii) Rearranging specific k-cycles also causes repetitions, e.g., (1)(2)(3) and (2)(3)(1) are the same three 1-cycles, or fixed points. There are c 1 !c 2 ! :::c n ! many such repetitions. Thus, dividing the number of rearrangements n!, with these numbers of repetitions we obtain the result. Now we define the cycle index of S n . Definition 2.1.8. Let C be as above. Then we define the cycle index (or cycle indicator) of S n to be the polynomial Z n =Z n (t 1 ;t 2 ;:::;t n ) = 1 n! X 2Sn t c1 1 t c2 2 :::t cn n : Example 2.1.9. The cycle index for S 3 isZ 3 = 1=6(t 3 1 + 3t 1 t 2 + 2t 3 ). This is reflected in the fact that jZ 3 j = 6 and that the elements of S 3 are (1)(2)(3); (1)(23); (13)(2); (12)(3); (123); (132): The coefficients of the terms in the cycle index tells how many such permutations S n has. In this case only one permutation of type (3; 0; 0), one being the coefficient oft 3 1 , three permutations of type (1; 1; 0), three being the coefficient oft 1 t 2 , and lastly two permutations of type (0; 0; 1), two being the coefficient of t 3 . Remark 2.1.10. Using Proposition 2.1.7 we see that Z n = 1 n! X 2Sn t c1 1 t c2 2 :::t cn n = 1 n! X c1+2c2+:::+ncn=n n! 1 c1 c 1 ! 2 c2 c 2 ! :::n cn c n ! t c1 1 t c2 2 :::t cn n : 9 The next theorem is quite helpful in utilization of the cycle index. Theorem 2.1.11 (Theorem 1.3.3 in Stanley (2011)). Summing the cycle indices for n 0, we get X n0 Z n x n =exp t 1 x +t 2 x 2 2 +t 3 x 3 3 +::: : Proof. First, we will expand the right hand side. exp t 1 x +t 2 x 2 2 +t 3 x 3 3 +::: = exp 0 @ X i1 t i x i i 1 A = Y i1 exp t i x i i = Y i1 X j0 t j i x i i j =j! = Y i1 X j0 x ij t j i i j j! = 0 @ X j0 x j t j 1 1 j j! 1 A 0 @ X j0 x 2j t j 2 2 j j! 1 A ::: 0 @ X j0 x kj t j k k j j! 1 A ::: Now, we will compare the coefficients of t c1 1 t c2 2 :::t cn n x n in both cases. In this last product, to form the term t c1 1 t c2 2 :::t cn n x n , we will have x c1 1 c1 c 1 ! x 2c2 2 c2 c 2 ! ::: x ncn 1 cn c n ! ; all the while, sum of the powers of x, c 1 + 2c 2 +::: +nc n must add up to n. Then the coefficient will become 1 1 c1 c 1 ! 2 c2 c 2 !:::n cn c n ! = 1 n! n! 1 c1 c 1 ! 2 c2 c 2 !:::n cn c n ! : Recalling from Remark 2.1.10 that this is exactly the coefficient within Z n , we are done. 10 2.1.2 Expected Value and Asymptotic Distribution of Number of k-Cycles Here we will provide an example to see an example of how useful the cycle index is. Recall that we are mainly interested in the number ofk-cycles for a uniformly randomly selected permutation. We will denote this random variable by C n;k . Throughout this dissertation, we use the same notation, C n;k , taken under different subsets of S n , which is a slight abuse of notation. Now we have E[C n;k ] = 1 n! X 2Sn c k (); where c k () is the number of k-cycles in . Recalling that Z n = 1 n! X 2Sn t c1 1 t c2 2 :::t cn n ; we get E[C n;k ] = @ @t k Z n (t 1 ;t 2 ;:::;t n )j ti=1 : Therefore, by Theorem 2.1.11 we get X n0 E[C n;k ]x n = @ @t k exp t 1 x +t 2 x 2 2 +t 3 x 3 3 +::: j ti=1 = x k k exp x + x 2 2 + x 3 3 +::: = x k k exp log(1x) 1 = x k k 1 1x = x k k X n0 x n : So we have E[C n;k ] = 1=k. The fact that larger cycles are less likely makes intuitive sense. Also, this result is also in accordance with the fact that the asymptotic distribution of C n;k is Poisson(1=k). 11 We will now present this asymptotic result as provided in Arratia and Tavare’s 1992 paper “The Cycle Structure of Random Permutations” [5]. Theorem 2.1.12 (Theorem 1 in Arratia (1992)). For k = 1; 2;:::, let C n;k be the number of k-cycles of a uniformly randomly selected permutation 2 S n . Then the process of cycle counts converges in distribution to a Poisson process onN with intensity k 1 . In other words, as n!1 (C n;1 ;C n;2 ;:::;C n;k ;:::)) (Z 1 ;Z 2 ;:::;Z k ;:::); where Z i , i = 1; 2;:::; are independent Poisson random variables with mean 1=i. There are various ways to prove Theorem 2.1.12. For example in [5] a proof that uses the method of moments is provided. In this paper they provide much more, including upper bounds to the total variation distance (which will be defined in Chapter 4) between the dependently joint C n;k and inde- pendently joint Poisson(k) variables. They also provide similar results for other scenarios, where only certain permutations with specified cycle structures are considered. Theorem 2.1.12 has a rather long history that goes about three centuries ago. The case k = 1, i.e., the number of fixed points having Poisson(1) asymptotic distribution was proven by Montmort (1708) [12], before Poisson was eve born! The earliest joint distributional result is in the works Goncharov (1942, 1944) [20] and [21]. For a more detailed discussion and a wider literature refer to Arratia and Tavare’s (2003) text, “Logarithmic Combinatorial Structures: A Probabilistic Approach” [4]. Our main result will be similar asymptotic results, but not on S n , but subsets of it that adheres to certain positional restrictions. In the next section we will provide the main form of these positional restrictions. 2.2 Restriction Matrix There are various types of restrictions imposed on S n that have been studied. Pattern avoiding permutations are amongst the most popular examples. Another approach is to focus attention to 12 permutations with specific cycle types. We will concentrate on positional restrictions throughout this thesis. The most direct way to indicate positional restrictions is through 0 1 matrices. There is a strong connection between 0 1 matrices with the corresponding permutations and Ferrer’s boards with the corresponding placements of non-attacking rooks. Diaconis, Graham and Holmes’ (2001) paper, titled “Statistical Problems Involving Permutations with Restricted Positions” [13] provides a great survey of connectionsbetweenpermutationswithrestrictedpositionsandmanydifferentfields, includingcomplex- ity theory, matching theory, rook polynomials and statistical tests for independence. Another reference for this connection is Hanlon’s 1996 paper “A Random Walk on the Rook Placements on a Ferrers Board” [22]. We will occasionally refer to these papers throughout the dissertation. We start by giving the definition of restriction matrices. Definition 2.2.1 (Restriction Matrix). Let M be an nn matrix with all entries 0 or 1. We refer to M as a restriction matrix if it determines which label of a permutation is allowed at what position. Label j is allowed at position i, if M i;j = 1 and not allowed otherwise. For a given restriction matrix M, we define S M n S n to be the set of permutations in accordance with M. As there are n 2 entries of M, there are 2 n 2 such possible matrices. Not only some of these matrices correspondtothesamesubsetofS n , butalsoasignificantportionofthemexpressthesamepermutations up to the order. Consider the following example. Example 2.2.2. Let A i be the restriction matrix with all 1’s, except the entry (i;i). Then S Ai n is the set of permutations without the 1-cycle, (i). In this case for anyi;j2f1; 2;:::;ng,S Ai n andS Aj n would be equivalent in the sense that we would obtain S Aj n if we switched the roles of i and j in S Ai n . There is not an exact formula to determine the number of essentially different cases. For a discussion refer to Riordan’s text, “An Introduction to Combinatorial Analysis”, p. 164 [31]. 13 IfM is the matrix of all 1’s, thenS M n =S n , i.e., there is no restriction. However if M is the matrix of all 1’s except the diagonal, then S M n corresponds to the permutations that work as the appropriate arrangements in the famous Rencontres Problem. Example 2.2.3 (Probleme des Rencontres). Let M nn be the following matrix 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 0 1 1 1 ::: 1 1 1 1 0 1 1 ::: 1 1 1 1 1 0 1 ::: 1 1 1 1 1 1 0 ::: 1 1 1 . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1 1 ::: 0 1 1 1 1 1 1 ::: 1 0 1 1 1 1 1 ::: 1 1 0 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 : This restriction matrix results in the permutations that have no fixed points. Imagine the scenario where n mathematicians, each with a hat, go to a restaurant, and on the way out they get their hats back in a random order. What is the probability that none of them gets their own hats? This famous problem is immediately translated into the problem “For a randomly selected permutation2S n , what is the chance that it will be in S M n ”? The answer is directly seen to be jS M n j n! : The trouble here is calculating the valuejS M j. The same problem was considered by Euler, who provided a nice recursive formula. Proposition 2.2.4 (p.60 in Riordan(2002)). LetD n be the number of permutations in S n with no fixed points. Then D n = (n 1) (D n1 +D n2 ): 14 Proof. First position might be any label except 1, so that leaves us with n 1 options. Suppose now (1) = i, then there are two possibilities for (i), either 1 or something other than both 1 and i. In case(i),wewouldhavea 2-cycle(oratransposition),namely((1 i),whichleavestheremainingn2labels and positions free. Then we would haveD n2 ways to make sure these have no fixed points. In case (ii) the allowed permutations that are those with labels 2; 3;:::;n and positions 1; 2;:::;k 1;k + 1;:::;n, for whichk cannot be mapped to 1 and 1 cannot be mapped tok and no other label cannot be mapped to itself. This is exactly D n1 , as we may think label k as the new label 1. Recursion follows. It is also easy to prove that D n =n! P n i=0 (1) i i! . Rearranging the terms, we get n! n X i=0 (1) i i! =n! n X i=0 (1) i i! = n X i=0 n i (1) i (ni)!: But this is exactly the application of the principle of inclusion and exclusion to the eventsf(i)6=ig for i2f1; 2;:::;ng. There are many interesting problems in combinatorics and probability theory that can be expressed using permutations with restricted positions. Rencontres was only one such example. Different versions of the Menages Problem can also be described by 0 1 restriction matrices as above. The sensed difficulty of these counting problems is inherent in the general set up of permutations with restricted positions as we will see in the next section. 2.2.1 Permanent In this section we will go over the concept of the permanent of a 0 1 matrix and its computation. To start, we present an alternative expression for S M n , the set of permutations in accordance with the restriction matrix M, 15 S M =f2S n : n Y i=1 M i(i) = 1g: Theformulaexplainsclearlythatfor tobeacorrespondingpermutationallassignmentsitproposes, (i;(i)) must be allowed. Then we immediately get jS M j = X 2Sn n Y i=1 M i(i) =Per(M): The first equation follows due to the fact that only permutations in accordance with M contribute to the sum, and they only contribute a factor of 1. The second equation is directly the definition of the permanent of a matrix. Recall the definition of the determinant of a matrix, Det(M) = X 2Sn n Y i=1 (1) i M i(i) : Simply put, permanent is the determinant without alternating powers of1. Despite these simi- larities, there is a polynomial time algorithm to calculate the determinant via Gaussian Elimination, whereas no such algorithm exists for permanent at the moment. Actually, finding such a formula would be a groundbreaking result in mathematics and computer science, as it would imply P =NP and much more! Calculating the permanent from its definition takes nn! many operations. Ryser, in 1963, provides an algorithm that brings the calculation to an exponential time,O(n2 n ) (see [33]). That is still the fastest exact algorithm to the day, up to the order. Further, in 1979, Leslie Gabriel Valiant proved that calculating the permanent of a 0 1 matrix is at least as difficult as a NP problem. In his paper “The Complexity of Computing the Permanent” [35] he presents the problem of calculating the permanent of a 0 1 matrix as the first example of a #P problem, defining #P as a complexity class. Crudely put, #P problems are counting versions of the decision, or existence problems of NP class. Example 2.2.5. The following are NP (decision) problems. 16 Is there any subset of a list of integers with sum being equal to zero? Is there any Hamiltonian cycle in a given graph with cost less than 100? These are their #P (counting) counterparts. How many subsets of a list of integers add up to zero? How many Hamiltonian cycles in a given graph have cost less than 100? Despite the fact that counting versions of NP problems are #P, the other direction is not always true. Example 2.2.6. Deciding on the existence of a perfect matching for a given bipartite graph is known to have a polynomial time solution (see Edmonds (1965) [17]). On the other hand, counting the number of perfect matchings in a given bipartite graph is shown to be #P-complete. Now that the difficulty establishing a polynomial time - exact algorithm is apparent, we will describe another approach to this problem. It starts by visualizing the 01 matrixM as the adjacency matrix of a bipartite graph. Then any perfect matching in this bipartite graph will correspond to a permutation in accordance with M. Using this one-to-one correspondence if we can count the perfect matchings in a given bipartite graph, then we would be done. But as we saw in Example 2.2.6 this was already a difficult problem. In 1986, Broder proposed to use Monte Carlo Markov Chain method for sampling perfect matchings, which in the same year was shown by Jerrum and Sinclair to be equivalent to provide a FPRAS (Fully Polynomial Randomized Approximation Scheme) to the permanent calculation problem. A FPRAS would work in polynomial time and for any level of approximation error provide a value within that error level of the true value of the permanent with high probability. Basically, this is one of the best alternatives to an exact polynomial algorithm. Broder’s proposed Markov chain was defined on perfect and near-perfect (if only two nodes are missed) matchings, and his method only worked when the near-perfect matchings did not outnumber 17 perfect matchings. In 1989, through this Markov Chain Jerrum and Sinclair provided the FPRAS for dense matrices and almost every random matrix. For more than a decade the problem for an arbitrary M endured, but in 2004 Jerrum, Sinclair and Vigoda solved the problem completely in their paper “A Polynomial-Time Approximation Algorithm for the Permanent of a Matrix with Nonnegative Entries” [27]. Diaconis et at 2001,[13] is another good reference for the permanent calculation problem, noting that the final solution of FPRAS did not exist by the time this paper was written. Now we turn our attention to specific types of restriction. Whenever the ones that appear at all rows oftherestrictionmatrixM areconsecutive,therestrictionisreferredtoasinterval restriction. Adetailed study on these restrictions and the random transposition walks on the corresponding permutations is Olena Bormashenko’s thesis [9] (2011). 2.3 Two-sided Restrictions An important special case of interval restriction is two-sided restriction. In terms of labels, two-sided restrictions might be expressed as a i (i)b i ; where each a i and b i are positive integer lower and upper bounds to the values (i) might take on. These permutations are used in coding theory, when the amount of potential displacement of characters that are transmitted is known. For example consider the bounds, im(i)i +m. Here we know that each character might be displaced an amount ofm unit above or below. For applications in coding theory, as well as the generating function for the number of corresponding permutations for m 6 refer to Klove’s 2009 paper, titled “Generating Functions for the Number of Permutations with Limited Displacement” [28]. Some non-symmetric versions are also studied in “On the Number of Certain Types 18 of Strongly Restricted Permutations”, by Baltic (2010) [6]. They provide a concise history of the two- sided restrictions and related problems, as well as provide a new method that creates linear recurrences to count the number of restricted permutations in certain cases. In this section we will provide a working example to observe the changes that occur in the cycle structures of permutations with restricted positions, and bijections to other combinatorial objects. This is the aforementioned symmetric situation, im(i)i +m, wheren m = 1. Consider the following restriction matrix, A n = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 1 1 0 0 ::: 0 0 0 1 1 1 0 ::: 0 0 0 0 1 1 1 ::: 0 0 0 0 0 1 1 ::: 0 0 0 . . . . . . . . . . . . . . . . . . . . . . . . 0 0 0 0 ::: 1 1 0 0 0 0 0 ::: 1 1 1 0 0 0 0 ::: 0 1 1 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 : Definition 2.3.1. Elements ofS An n are called Fibonacci permutations. The reason for this naming will be understood below. For these permutations the restriction condition is i 1 (i) i + 1, i.e., we only allow one unit of displacement for each position. Expanding the permanent along the first row, we see that the submatrix obtained by deleting the first row and column is a matrix of tridiagonal 1’s, similar toA, but of dimension n 1, i.e., A n1 . The submatrix obtained by deleting the first row and second column of A n may also seen as tridiagonal, but after the cancellations in the permanent its dimension falls down to n 2. This implies a familiar recursion, Per(A n ) =Per(A n1 ) +Per(A n2 ): 19 Noting that Per(A 1 ) = 1;Per(A 2 ) = 2, we get Per(A n ) = F n+1 , where F i is the ith Fibonacci number. As we know that F n = 1 p 5 (' n ^ ' n ); where ' = 1 + p 5 2 ; ^ ' = 1 p 5 2 ; we have a direct formula forjS An n j. There is more. Due to the nature of restriction, Fibonacci permutations may only have 1-cycles or 2-cycles. For a cycle larger than a 1-cycle or 2-cycle, a label i has to be mapped to i 1 ori + 1 and, WLOG, i + 1 can only be mapped to i + 2. But as this process never stops, we cannot form the cycle. There are many other combinatorial objects that are equivalent in nature to Fibonacci permutations. Proposition 2.3.2 (Proposition 2.1 in Diaconis et al (2001)). The set of Fibonacci permutations on n letters, S An n is in one-to-one correspondence with: Compositions of n with all parts equal to one or two Matchings in an n-path Subsets of [n 1] =f1; 2;:::;n 1g with no consecutive elements Binary (n 1)-tuples without two consecutive ones. We are especially interested in the first set of objects, compositions of n with all parts equal to one or two, as they will show up in Chapter 5, during our discussion of colored compositions. Let us see Proposition 2.3.2 on the example of S A4 4 . Note that F 5 = 5, and the five Fibonacci permutations are (1)(2)(3)(4); (1)(2)(34); (1)(23)(4); (12)(3)(4); (12)(34): 20 Then we have the following corresponding objects. (1) Compositions of 4, with all parts equal to one or two: 1 + 1 + 1 + 1; 2 + 1 + 1; 1 + 2 + 1; 1 + 1 + 2; 2 + 2 (2) Matchings in a 4-path ! (1)(2)(3)(4) ! (12)(3)(4) ! (1)(23)(4) ! (1)(2)(34) ! (12)(34) (3) Subsets of [3] =f1; 2; 3g with no consecutive elements ;;f1g;f2g;f3g;f1; 3g: (4) Binary 3-tuples without two consecutive ones 000; 100; 010; 001; 101: RecallingthatFibonaccipermutationsonlyhave 1-cycles(fixedpoints)and 2-cycles(transpositions), we adopt the aforementioned slight abuse of notation and let C n;1 (respectively, C n;2 ) be the number of fixed points (resp., transpositions) of a Fibonacci permutations that is selected uniformly at random. These two random variables satisfy the condition C n;1 + 2C n;2 =n; asthesearetheonlycycletypesthatallowed. Thisallowsthecomfortablecalculationofprobabilities for C n;1 given the probabilities of C n;2 , or vice versa. It will turn out that the distribution of C n;2 is 21 asymptotically Normal, which, in turn, implies that the distribution of C n;1 is asymptotically Normal. We present the following proposition without proving. For more detail refer to [13]. Proposition 2.3.3 (Proposition 2.4 in Diaconis et al (2001)). Let 2 S An n be a randomly selected Fibonacci permutation. Then P (C n;2 =k) = nk k F n+1 ; 0kb n 2 c; E[C n;2 ] = n( p 5 1) 2 p 5 + ^ ' 5 +O(n ^ ' 2n ); Var n (C n;2 ) = n 5 p 5 +O(1); P C n;2 E[C n;2 ] p Var n (C n;2 ) x ! ) 1 p 2 Z 1 e t 2 2 dt: 2.4 One-sided Restrictions Another important type of restriction is one-sided restriction. This time, in terms of labels, the restrictions might be expressed as b i (i); only in terms of lower bounds. We choose to work with lower bounds instead of upper bounds without loss of generality. If we were to work with upper bounds, then the corresponding permutations would be equivalent to lower bounded versions, up to the symmetry of labels. 22 In terms of restriction matrices, one-sided restrictions correspond to matrices with having zeroes only at their lower-diagonal part. Hanlon, in his 1996 paper “A Random Walk on the Rook Placements on a Ferrers Board” provides a general survey of permutations with one-sided restrictions and studies the random transposition walk on these permutations [22]. We will follow Hanlon’s suit and refer to these permutations as b-regular. Definition 2.4.1 (b-regular permutations). Letb = [b 1 b 2 :::b n ] be a vector of non-decreasing, positive integers such that b i 2f1; 2;:::;ng. Then we call b a restriction vector if we impose the condition b i (i) on 2 S n , and refer to such permutations as b-regular. We will notate the set of b-regular permutations as S b n . One immediate result is that b i i in order for the restriction to be nontrivial. Also, the choice of b i to be non-decreasing is WLOG, as otherwise we would again have equivalent permutations up to the symmetry. Thinking in terms of the corresponding restriction matrix, the entries of ones on the ith row start at column b i . Example 2.4.2. Consider the following restriction vector b = [1 2 2 3]. The corresponding restriction matrix is M b = 2 6 6 6 6 6 6 6 6 6 6 4 1 1 1 1 0 1 1 1 0 1 1 1 0 0 1 1 3 7 7 7 7 7 7 7 7 7 7 5 : Example 2.4.3. Another example is what we will call b (2) -regular permutations. For n = 5, the restriction vectors is b (2) = [1 1 2 3 4], and the corresponding restriction matrix is 23 M b = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 1 1 1 0 0 0 1 1 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 : Again, we see that the numbers b (2) i correspond to where the 1’s start at each row. b (2) restriction vector has two initial 1’s and then increase consecutively tilln1. We will also define the generalization of these, b (r) -regular permutations. The number of permutations, in accordance with the restriction vector b, as well as the generating functions for the number of cycles and number of inversions are well-known and are provided in Hanlon’s paper (1996) [22], while himself referring to the previous work of Goldman, Joichi and White, “Rook Theory” I-II-III-IV [19]. Proposition 2.4.4. [Proposition 2.1 in Hanlon 1996] Let b be a restriction vector, and for any permutation , c() and i() be number of cycles and inversions of . Then (a) jS b n j = Q n i=1 (1 + (ib i )), (b) P 2S b n q c() = Q n i=1 (q + (ib i )), (c) P 2S b n q i() = Q n i=1 1q 1+ib i 1q . We are particularly interested in part (a), namely the number of b-regular permutations jS b n j = n Y i=1 (1 + (ib i )); (2.4.1) which will be used to calculate the first two moments of the number of fixed points for a randomly selected b-regular permutation. 24 2.4.1 Moment Calculations for Fixed Points Differently than the classical unrestricted case, we first need to identify which cycles are allowed and which are not due to the restrictions. Depending on the restrictions the distributions of the number of k-cycles might change critically. For example, in the case of M being the matrix with 1’s everywhere except the main diagonal, there could be no fixed points, where without the restriction there would be around one fixed point on average. Whatever the imposed restriction might be, we cannot ignore these changes. One way to study the problem is through expressing the number of k-cycles as a sum of indicators, i.e., C n;k () = X c is an allowed k-cycle 1fc is a cycle of g: For a given vectorb we will calculate the expected value and variance of the number of fixed points, C n;1 (), where is a randomly chosen b-regular permutation. Using the indicator representation of the number of fixed points, C n;1 () = P n i=1 1f(i) = ig, we need to calculate probabilities of the form P ((i) = i), which is equivalent to counting number of b-regular permutations with “(i) = i” happening. We calculate the permanents of the matrices with the corresponding conditioning. For instance, consider the restriction vector b = [1; 1; 2; 4; 4]. The corresponding restriction matrix is M b = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 1 0 0 0 1 1 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 : If we were to condition on the event thatf(2) = 2g, then as we prevent (2) to have any value other than 2 and vice versa, the corresponding matrix would have a 1 only at the position (2; 2) along the column and row number two: 25 M b;(2) = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 1 0 1 1 1 0 1 0 0 0 0 0 1 1 1 0 0 0 1 1 0 0 0 1 1 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 : Clearly, the permanent ofM b;(2) is equal to the permanent of the smaller matrix where the row and column two are deleted: M b;(2) = 2 6 6 6 6 6 6 6 6 6 6 4 1 1 1 1 0 1 1 1 0 0 1 1 0 0 1 1 3 7 7 7 7 7 7 7 7 7 7 5 : The critical observation here is that this new matrix M b;(2) corresponds to a new and smaller restriction vector. We can imagine this vector as a restriction to be imposed on the permutations of the four numbersf1; 3; 4; 5g. Permanent of this smaller matrix is equal to the formula in Equation 2.4.1 applied to the new restriction vector, b (2) = [1; 2; 3; 3]. To understand how the conditioned event “(2) = 2” determines the corresponding vector b (2) , we observe the fact that deleting the column and row 2 would decrease the number of ones any row that would haveb i 2. Otherwise they are not affected. This immediately implies the following proposition. Proposition 2.4.5. The number of b-regular permutations with (i) =i equals to the number of b (i) - regular permutations whereb (i) is the vector obtained by erasingith entry ofb and decreasing the entries j >i given that b (i) j >i, otherwise leaving them as they are. Proposition leads to our result on the calculation of the expected number of fixed points and its variance. 26 Theorem 2.4.6 (Proposition 2.2 in Ozel 2017). Let be a randomly chosen b-regular permutation. Then E[C n;1 ()] = 1 jS b n j n X k=1 jS b;k n j and Var(C n;1 ()) = 1 jS b n j n X k=1 h S b;k n (S b n S b;k n ) i + 2 X i<j h S b;i;j n S b;i n S b;j n i ; where S b;k n = k1 Q j=1 (1 +jb j ) n Q j=k+1 (j (b j 1(b j >k))) and S b;i;j n = i1 Y m=1 1 +mb m j1 Y m=i+1 1 +m (b m+1 1(b m+1 >i)) n Y m=j+1 1 +m (b m+2 1(b m+2 >i)) 1(b m+2 >j)) : Proof. S b n was the number of b-regular permutations, given in the formula (2.4.1). By Remark 2.4.1 the number of b-regular permutations where k is a fixed point isjS b;k n j and the number of b-regular permutations with i and j are fixed points isjS b;i;j n j, using the remark twice. The results follow from the linearity of expected value and covariance formula for the variance of sum of indicators. Example 2.4.7. Letb + = [1 2 2 :::; 2], then for a uniformly randomly selectedb + -regular permutation , we have jS b + n j = n Y i=1 (1 +ib i ) = (1 + 1 1) n Y i=2 (1 +i 2) = 1 1 2 3 ::: (n 1) = (n 1)!: Now we will calculate the values for S b + ;k n . For k = 1, S b + ;1 n = n Y j=2 (jb j + 1(2> 1)) = n Y j=2 (jb j + 1) = 1 1 2 3 ::: (n 1) = (n 1)!: On the other hand, for k 2, 27 S b + ;k n = 1 1 2 3 ::: (n 2) = (n 2)!: Combining all, E[C n;1 ()] = (n 1)! + (n 2)! +::: + (n 2)! (n 1)! = (n 1)! + (n 1)(n 2)! (n 1)! = 1 + 1 = 2: Noticing that this restriction matrix has the condition b i 2 for each i 2, we see that label 1 can only be positioned at position 1. This makes (1) = 1 a certain fixed point. The remaining positions, 2; 3;:::;n, are only mapped to themselves, resulting in (n 1)! unrestricted permutations on n 1 letters. Recalling the result in Section 2.1.2, namely, E[C n;k ] = 1=k, and thus, for k = 1, E[C n;1 ] = 1=1 = 1. Combining this averaged one fixed point with the certain fixed points (1) = 1, we confirm the answer should be 2. Remark 2.4.8. This method is applicable to other cycle types as well, granted that we know which of the cycles of that type are allowed. For b-regular permutations, any coordinate might be a fixed point by the assumptionb i i. But for transpositions and bigger cycles this identification is necessary. Furthermore, conditioning on a k-cycle will require k indicators to be subtracted and thus calculating higher moments will become more and more expensive. We will sacrifice generality for utility and use cycle index for a specific family of b-regular permutations. In the next chapter we will focus on a specific type of restriction vectors and identify bijections betweenthecorrespondingpermutationsandothercombinatorialobjects, pavingthewaytoestablishing the asymptotic distribution for C n;k . 28 Chapter 3 b (r) -Regular Permutations and Compositions In this chapter we will introduce the main objects of this dissertation’s interest, b (r) -regular permu- tations. Next, we will establish an important bijection between S b (2) n and the set of the compositions of n. This bijection translates the cycle structure of permutations to part structure of compositions, which will lead to the answers of many questions regarding compositions. Then we will provide a brief survey on results related to compositions, which will be followed by our investigation into b (3) -regular permu- tations and colored compositions. We are indebted to Professor Jason Fulman 1 for their suggestion to use the bijective approach and the calculations for the cycle index. 3.1 b (r) -Regular Permutations Letb (r) be the restriction vector that starts withr many 1’s, then increase consecutively tillnr+1. The permutations corresponding to this restriction vector are calledb (r) -regular permutations. Note that this restriction may be expressed as (i)ir + 1 for ir. For i<r, there is no non-trivial restriction as we only impose (i) 1. 1 University of Southern California, Department of Mathematics 29 The restriction matrices corresponding to these restriction vectors have all upper triangular 1’s,r1 lower sub diagonals of 1’s, and 0’s everywhere else. So, moving from r = 0 to r = n, they provide a means to observe how the cycle structure evolves from the trivial case r = 0, leading to the single permutation (1)(2)::: (n) to the whole symmetric group S n . Our long-term research objective is to observe this evolution of cycle structure. Using Proposition 2.4.4, we can easily count the number of b (r) -regular permutations, jS b (r) n j = n Y i=1 (1 +ib (r) i ) =r!r nr : For the case r = 2, this gives 2! 2 n2 = 2 n1 . There are many combinatorial objects with this cardinality, some of which share peculiar properties with b (2) -regular permutations. Hanlon (1996) [22] uses the fact that there are 2 n1 vertices of a hypercube to establish a correspondence between the random transposition walk over the b (2) -regular permutations and the random walk on the hypercube. Before proceeding to properties ofb (2) -regular permutations, we present the following calculation for the expected number of fixed points for b (r) -regular permutations using Theorem 2.4.6. Proposition 3.1.1. Let be a uniformly randomly selectedb (r) -regular permutation. Then the expected number of fixed points is E[C n;1 ()] = 2 + n 4r + 2 r 1 1 r r1 : Here we assume n large enough, i.e., n 2r 1. 30 Proof. Recalling that S b;k n in Theorem 2.4.6 were the number of b-regular permutations where kth position is a fixed point, i.e., (k) =k. Checking the formulas for S b (r) ;1 n ;S b (r) ;2 n ;:::;S b (r) ;n n , we observe a symmetric pattern, namely jS b (r) ;1 n j =jS b (r) ;n n j jS b (r) ;2 n j =jS b (r) ;n1 n j . . . jS b (r) ;r1 n j =jS b (r) ;nr+2 n j jS b (r) ;r n j =jS b (r) ;r+1 n j =::: =jS b (r) ;nr+1 n j: We calculatejS b (r) ;1 n j to be (r 1)!r nr , and further, as we progress along the indices of fixed points, one term of r is replaced by one term of (r 1), i.e., for j = 1; 2;:::;r, jS b (r) ;j n j = r 1 r jS b (r) ;j1 n j: Likewise, for j =n;n 1;:::;nr + 1 jS b (r) ;j n j = r 1 r jS b (r) ;j1 n j: The n 2r + 2 terms in between are equal to each other. From E[C n;1 ()] = 1 jS (r) b j n X k=1 jS b (r) ;k n j; we divide each term by S b (r) ;k n with S b (r) n =r!r nr to obtain the sum r r1 r r + (r 1)r r2 r r +::: + (r 1) r2 r r r + (r 1) r1 r r +::: + (r 1) r1 r r | {z } n2r+2 terms +::: + (r 1)r r2 r r + r r1 r r : 31 Beforedividingwithr r ,wewillfirstaddupther termsfromeachend,i.e.,indices 1;:::;r;nr+1;:::;n. This yields the sum 2 r r1 + (r 1)r r2 + (r 1) 2 r r3 +::: + (r 1) r2 r + (r 1) r1 = 2 (r r (r 1) r ): Adding the remaining n 2r midmost terms, we get 2 (r r (r 1) r ) + (n 2r)(r 1) r1 = 2r r + (n 2r 2r + 2)(r 1) r1 : Now, dividing by the denominator of r r , we get E[C n;1 ()] = 2r r + (n 2r 2r + 2)(r 1) r1 r r = 2 + n 4r + 2 r 1 1 r r1 as desired. Forr = 2, this corresponds toE[C n;1 ()] = n+2 4 . We will calculate this value and much more, using another approach as we noted the inflexibility of the permanent method in Remark 2.4.8. 3.1.1 b (2) -Regular Permutations and k-Cycles Our main result will be the asymptotic distribution of k-cycles of b (2) -regular permutations. Thus we would like to analyze these permutations and their k-cycles in more detail. First we identify the unique characteristic of the cycles of b (2) -regular permutations with the following proposition. Proposition 3.1.2. Let be a b (2) -regular permutation, then any k-cycle of will be of the form (m k m k1 m k2 :::m 2 m 1 ) = (m 1 m k m k1 :::m 3 m 2 ); (3.1.1) when written in the cycle notation. Here m k is the greatest label within the cycle and m i are increasing, consecutive numbers. 32 Proof. Recall that the b (2) restriction may be stated as (i) > i 1 for each i2f1; 2;:::;ng. This restriction implies that each (i) has to be mapped to at least one coordinate below. Say m k is the largest position within a k-cycle, then it has to be mapped to m k 1 unless it is a fixed point, i.e., (m k ) =m k and thus k = 1. Similarly, m k 1 has to be mapped to m k 2, and following iteratively we reach down to m 1 , which has to be mapped to m k , which is allowed as there are no upper bound restrictions. Example3.1.3. ToobserveProposition3.1.2,considerthesetofb (2) -regularpermutationsonf1; 2; 3; 4g. There are 2 41 = 8 of these permutations and they are as follows when written in cycle notation. (1)(2)(3)(4); (12)(3)(4); (1)(23)(4); (1)(2)(34); (12)(34); (132)(4); (1)(243); (1432) It is clear that each k-cycle is of the form (m 1 m k m k1 :::m 3 m 2 ), where m i =m k k +i. Now that we have a picture of a generic k-cycle we can list them all possible k-cycles. Corollary 3.1.4. Let 2 S b (2) n , then might potentially have the following n k + 1 k-cycles, c k 1 ;c k 2 ;:::;c k nk+1 , each cycle is of the form Equation 3.1.1. Written in the cycle notation, c k i has i as its smallest element. Proof. Proposition 3.1.2 already established the form the allowed k-cycles. We only need to decide the smallest (or the largest) number in the cycle. As the smallest number can at least be 1 and at most be nk + 1, there are nk + 1 many different potential k-cycles. Naming these cycles c k i , where i is the smallest number, we are done. Example 3.1.5. Consider n = 5. Then the potential 2-cycles are (12); (23); (34); (45) 33 3-cycles are (132); (243); (354) 4-cycles are (1432); (2543) and the only possible 5-cycle is (15432): In the next section we will briefly introduce compositions ofn and establish the bijection we will use between b (2) -regular permutations and compositions. 3.2 Compositions We start with the definition. Definition 3.2.1. A composition of a positive integern is any ordered list = 1 2 ::: m of positive integers such that P m i=1 i =n. We will also use the notation = 1 + 2 +::: + m . i are called parts of the composition,par() =m counts the number of parts of a given composition . n is also referred to as the order of . There are 2 n1 compositions of n, which is easy to see if we consider n = 1 + 1 +::: + 1: 34 There aren 1’s and thusn1 plus signs in this expression. For each plus sign, there are two choices, leading to a total of 2 n1 arrangements and thus 2 n1 different compositions. The more classical approach is to consider the path with n edges. o o Example 3.2.2. In this case we have 10 edges and 11 nodes, but the two end nodes do not contribute to the selections. Then any selection of the remaining 9 nodes will correspond to a specific composition of 10, of which there are 2 9 . For example, o x x x o corresponds to the composition 2 + 4 + 2 + 2, whereas o x x x o describes 1 + 1 + 6 + 2. This approach makes the following observation easy to prove. Proposition 3.2.3. The number of compositions of n with exactly m parts is n1 m1 . Proof. As noted above, there aren1 nodes to select from and to havem parts we needm1 selections. There are exactly n1 m1 ways to do this. A great reference for compositions is the text by Heubach and Mansour, “Combinatorics of Compo- sitions and Words” (2009) [23]. Not only they include a comprehensive overview of techniques that are used in the analysis of compositions and words, they also provide a vast literature review. Recall the Fibonacci permutations in Chapter 2, more specifically, Proposition 2.3.2. We noted in this proposition that there is a one-to-one correspondence between these permutations and compositions with all parts equal to one or two. Such compositions with various restrictions are studied in depth in Heubach and Mansour (2009). This specific case analyzed is Example 2.9. and Theorem 3.10. 35 3.2.1 The Bijection Inthissectionwedescribethebijectionbetweenthesetofb (2) -regularpermutationsandcompositions of n. To describe the bijection we need the concept of a record position. Definition 3.2.4. Let 2S n be any permutation, then i2f1; 2;:::;ng is called a record position of , if (i)>(j) for all j <i. Further, (i) is called a record value. Example 3.2.5. Consider = 1 2 3 4 5 2 1 5 3 4 ,i = 1 andi = 3 are the record positions of. Thus,(1) = 2 and (3) = 5 are the record values. By this definition,i = 1 is always a record position and this is inherently equivalent to the fact that every permutation has at least one cycle. Now we present the bijection. Proposition 3.2.6. [Lemma 2.5 in Ozel (2017)] There is a bijection between the set of b (2) -regular permutations and the set of compositions of n, which matches the cycle sizes of permutations to part sizes compositions. Proof. Let2S b (2) n andconsidertherecordpositionsof. Startingati = 1, thefirstrecordposition, we proceed till the next record position to construct the first part of the composition. Similarly, starting from any record position we proceed till the next record position after it. We will prove that this mapping is one-to-one and as the number of compositions of n is 2 n1 as well, the mapping will indeed be a bijection. By Proposition 3.1.2, we note that each cycle of a b (2) -regular permutation is in decreasing order. This implies that any record position will start a new cycle that will proceed till the next record position. This is as a record position is the greatest value till the next record position, thus the cycle it belongs to will start with it, and it will end with the next record position and a new cycle will begin. Let us assume we have two permutations 1 ; 2 2S b (2) n , and further, they are mapped to the same composition ofn. By the definition of our mapping, these permutations have the same record positions, and thus, the same cycle structure. The record values, too, must be the same, otherwise the permutation 36 with the higher record value will cease one of its latter record positions from being a record position. 1 and 2 have cycles of identical length, with identical maximum value, so by Proposition 3.1.2 they have the same cycles, or, equivalently, 1 = 2 . This proves that this mapping is one to one and therefore a bijection. Lastly, by the construction of the mapping and Proposition 3.1.2 k-cycle always corresponds to a k-part as desired. Example 3.2.7. Recall that the record positions of = 1 2 3 4 5 2 1 5 3 4 , were i = 1 and i = 3. Noting that is indeed b (2) -regular, the corresponding composition of 5 is 2 + 3. Writing in cycle notation helps with the visualization = (2 1)(5 4 3): has a 2-cycle followed by a 3-cycle. The fact that this bijection maps k-cycles to k-parts is very important. Not only it will help us construct the cycle index, further, it will translate results about cycles of permutations into parts of compositions. 3.2.2 Some Results Regarding Parts of Compositions Chapter8ofHeubachandMansour(2009)[23]isagreatreferencefortheasymptoticsofcomposition statistics. In this section we will present some results on parts of compositions, without proofs, which can immediately be translated to cycles of b (2) -regular permutations using our bijection. 37 Example 3.2.8 (Example 8.9 & 8.15 in Heubach and Mansour (2009)). Let be a uniformly ran- domly selected composition of n, and Par() be the number of parts of . Recall that there are 2 n1 compositions and n1 m1 compositions with m parts. Then the expected number of parts of is E[Par()] = 1 2 n1 n X m=1 m n 1 m 1 = 1 2 n1 " n X m=1 (m 1)! (n 1)! (nm)! (m 1)! + n X m=1 n 1 m 1 # = 1 2 n1 " (n 1) n X m=2 n 2 m 2 + 2 n1 # = 1 2 n1 (n 1) 2 n2 + 2 n1 = n 1 2 + 1 = n + 1 2 : Further, using the probability generating function for number of parts, it is possible to prove Var(Par()) = n 1 4 : These two moments are immediately translate to the total number of cycles for a uniformly randomly selected b (2) -regular permutations. Let C n =C n;1 +C n;2 +::: +C n;n , then E[C n ] = n + 1 2 Var(C n ) = n 1 4 : Another result is about the length of the largest cycle. 38 Proposition 3.2.9 (Theorem 8.39 in Heubach and Mansour (2009)). Let L n;max () denote the size of a maximal part of a uniformly random composition . Then E[L n;max ()] log 2 (n) + 0:3327461773 +d(log 2 (n)); where d(:) is a periodic function that has period one, mean zero, and small amplitude. The next result is about the number of distinct part sizes of a random composition, which, again, can now be understood as the number of distinct cycle sizes of a random b (2) -regular permutation. Theorem 3.2.10 (p.520 in Hitczenko and Stengle (2000) [24]). Let be a composition ofn, andD n () be the number of distinct part sizes of . Then E[D n ] = log 2 (n) + ln(2) 3 2 +g(log 2 (n)) +o(1); where 0:57721 is the Euler-Mascheroniconstant, andg is a mean-zero function of period 1, satisfying jgj 0:0000016. Hitczenko has more results concerning various asymptotic properties of compositions. Another related result is on the number of k-parts of a random composition, where k is also chosen randomly from possible values of 1kn. Specifically they prove that, the probability that this numberA (m) n is equal to some numberj is asymptotically oscillating around 1 m ln(n) . For more detail refer to Hitczenko and Savage (2004) [25]. 3.3 Cycle Index of b (2) -Regular Permutations In this section we will establish the cycle index for b (2) -regular permutations. We will use the following Lemma in the calculations for cycle index. 39 Lemma 3.3.1. 2 + 1 X n=0 u n+1 2 n X 2S b (2) n Q i x ni() i ( P n i ())! = 2 1 Y i=1 e xi( u 2 ) i ; where n i () is the number of i-cycles of . Proof. By Lemma 3.2.6 we know cycles of 2S b (2) s n corresponds to parts of the mapped composition. This implies the number of b (2) -regular permutations with n i i-cycles is ( P i ni)! Q i ni! , under the assumption that P i in i =n. Therefore 1 + 1 X n=0 u n+1 X 2S b (2) n Q i x ni() i ( P i n i ())! = 1 Y i=1 1 X ni=0 (x i u i ) ni n i ! = 1 Y i=1 e xiu i Replacing u by u=2 and multiplying through by 2 gives the desired equation. Now we derive the cycle index for b (2) -regular permutations. Theorem 3.3.2. The cycle index for S b (2) n is given by 1 X n=0 u n+1 2 n X 2S b (2) n Y i x ni() i = 2 X j1 0 @ X i1 x i u 2 i 1 A j : Proof. In the Lemma 3.3.1 we replace each x i with tx i , we obtain 2 + 1 X n=0 u n+1 2 n X 2S b (2) n P ni()=j t j Q i x ni() i j! = 2 1 Y i=1 e txi( u 2 ) i Differentiating k times with respect to t and then setting t = 0 results 40 1 X n=0 u n+1 2 n X 2S b (2) n P ni()=j Y i x ni() i = 2 0 @ X i1 x i u 2 i 1 A j : Summing this equation over all j we obtain the desired equation, and our cycle index. Example3.3.3. Recallthatwecalculatedtheexpectednumberoffixedpointsforarandomb (2) -regular permutation using the permanent method to be n+2 4 . We now use the cycle index in Theorem 3.3.2 to make the same calculation. By setting x 1 =x and x i = 1 for i> 1, we obtain the generating function for the number of fixed points, 2 X j1 0 @ ux 2 + X i2 u 2 i 1 A j : Using formal power series, this can be rewritten as 2 2(2u) (2u)(2ux)u 2 1 : Taking its derivative with respect to x and setting x = 0 results in the generating function for the first moments, 4u(2u) 2 (4(1u)) 2 : Expanding this power series, we get 41 4u(2u) 2 (4(1u)) 2 = u 4 2u 1u 2 = u 4 1 + 1 1u 2 = u 4 0 @ 1 + 2 X n0 u n + X n0 (1 +n)u n 1 A = u 4 0 @ 1 + X n0 (n + 3)u n 1 A = u 4 + X n1 n + 2 4 u n : The general term of the power series, n+2 4 , is the desired quantity. Similarly, we can calculate any moment of k-cycles, by setting x k = x and other x i = 1 and differentiating as many times as necessary to obtain the desired degree of moment. This shows the generality and flexibility we obtain using the cycle index rather than the permanent method. Theorem 3.3.4. [Theorem 2.9 in Ozel (2017)] Let C n;k be the number of k-cycles of a uniformly randomly selected b (2) -regular permutation. Then n;k :=E[C n;k ] = nk + 3 2 k+1 ; and E[C n;k (C n;k 1)] = (n + 2 2k)(n + 7 2k) 4 k+1 ; combining we get 2 n;k :=Var(C n;k ) = (2 k+1 2k + 3)n + 3k(k 4) + (3k)2 k+1 + 5 4 k+1 : 42 Proof. Proceeding as in Example 3.3.3, we set x k =x and all other x i = 1. Differentiating once gives the expected value, differentiating twice gives the second factorial moment. Adding the correct amount of the first moment gives the variance. We now present corresponding calculations. Recall the cycle index was G(u;x 1 ;x 2 ;:::) := 2 X j1 0 @ X i1 x i u 2 i 1 A j : Now we let x k =x and x i = 1 for all i6=k as we are only interested in moments of k-cycles. Then this new generating function is G k (u;x) = 2 X j1 0 @ x u 2 k + X i1 u 2 i u 2 k 1 A j : We will reorganize the above expression using the properties of formal power series. G k (u;x) = 2 X j1 0 @ (x 1) u 2 k + X i1 u 2 i 1 A j = 2 X j1 (x 1) u 2 k + 1 1u=2 1 j = 2 X j1 (x 1) u k 2 k + u 2u j = 2 X j1 (x 1) (2u)u k +u 2 k 2 k (2u) j = 2 0 @ 1 1 (x1)(2u)u k +u2 k 2 k (2u) 1 1 A Reorganizing the rational expression, we obtain G k (u;x) = 2 2 k (2u) 2 k+1 (1u) (x 1) (2u)u k 1 ; 43 which may also be seen as a factorial moment generating function for C n;k . To get the expected value, we differentiate once and set x = 1. d dx G k (u;x) = 2 (2u)u k 2 k (2u) (2 k+1 (1u) (x 1) (2u)u k ) 2 ! = 2 k+1 (2u) 2 u k (2 k+1 (1u) (x 1) (2u)u k ) 2 Then setting x = 1, we get G ` k (u) = 2 k+1 (2u) 2 u k (2 k+1 (1u)) 2 = u k 2 k+1 2u 1u 2 = u k 2 k+1 1 + 1 1u 2 = u k 2 k+1 1 + 2 1 1u + 1 1u 2 ! = u k 2 k+1 0 @ 1 + 2 X n0 u n + X n0 (1 +n)u n 1 A = u k 2 k+1 0 @ 1 + X n0 (n + 3)u n 1 A = u k 2 k+1 + X n0 n + 3 2 k+1 u n+k = u k 2 k+1 + X nk nk + 3 2 k+1 u n : From the general term we get n;k =E[C n;k ] = nk + 3 2 k+1 ; (3.3.1) 44 which gives n+2 4 when substituted k = 1 as we calculated before. Now we would like to get the second factorial moment in order to calculate the variance. We take one more derivative of G k (u;x) and get d 2 dx 2 G k (u;x) = 2 k+1 (2u)u k 2 (2u)u k (2 k+1 (1u) (x 1) (2u)u k ) 3 = 2 k+2 (2u) 3 u 2k (2 k+1 (1u) (x 1) (2u)u k ) 3 : Setting x = 1, we get G \ k (u) = 2 k+2 (2u) 3 u 2k (2 k+1 (1u)) 3 = u 2k 2 2k+1 2u 1u 3 = u 2k 2 2k+1 1 + 1 1u 3 = u 2k 2 2k+1 1 + 3 1 1u + 3 1 1u 2 + 1 1u 3 ! = u 2k 2 2k+1 0 @ 1 + 3 X n0 u n + 3 X n0 (n + 1)u n + X n0 (n + 1)(n + 2) 2 u n 1 A = u 2k 2 2k+1 0 @ 1 + X n0 (n + 7)(n + 2) 2 u n 1 A = u 2k 2 2k+1 + X n0 (n + 2)(n + 7) 2 2k+2 u n+2k = u 2k 2 2k+1 + X n2k (n 2k + 2)(n 2k + 7) 2 2k+2 u n : From the general term we get E[C n;k (C n;k 1)] = (n 2k + 2)(n 2k + 7) 4 k+1 : (3.3.2) Combining Equation 3.3.1 and Equation 3.3.2, we get 45 2 n;k =Var(C n;k ) =E[C 2 n;k ]E[C n;k ] 2 =E[C n;k (C n;k 1)] +E[C n;k ]E[C n;k ] 2 = (n 2k + 2)(n 2k + 7) 4 k+1 + nk + 3 2 k+1 nk + 3 2 k+1 2 = (2 k+1 2k + 3)n + 3k(k 4) + (3k)2 k+1 + 5 4 k+1 as desired. Theorem 3.3.4 provides with the first two moments of the number of k-cycles for a uniformly ran- domly selected b (2) -regular permutation, which is a crucial ingredient for the central limit theorem we would like to establish. 46 Chapter 4 k-Cycle Indicators, m-Dependence and the Main Result In this chapter the number of k-cycles of a permutation , C n;k (), will be studied in more detail, specifically when is a b (2) -regular permutation selected uniformly at random. Recall that one way to represent the number of k-cycles was through indicators: C n;k () = X c is an allowed k-cycle 1fc is a k-cycle of g: (4.0.1) The conditional index in the sum 4.0.1 for a random permutation2S M n adhering to the restriction defined by the matrix M, expresses the need for identifying which k-cycles are allowed in this restric- tion. Recall that we addressed this identification for b (2) restriction in Section 3.1.1. The next step is to establish the local dependence structure between these indicator random variables for a randomly selected b (2) -regular permutation. There is a rather nice pattern of local dependence that is deter- mined by some form of distance between different cycles. Correspondingly distant cycles will be called p-separated. This form of local dependence is a weaker version of what is called m-dependence in the literature. Central limit theorems with m-dependence will be mentioned. Fortunately m-dependence, which is reasonably more difficult to prove will not be necessary to proceed. Stein’s method provides great distributional approximations for locally dependent random variables. The main result of this thesis will be established using such a result, Theorem 2.7 presented in [10]. 47 4.1 k-cycle Indicators of b (2) -regular Permutations In Proposition 3.1.2 and Corollary 3.1.4 we identified the potential k-cycles ofb (2) -regular permuta- tions. We denote the indicators of the corresponding cycles as follows I k i = 1fc k i is a k-cycle of g: Now we will calculate the probabilities of the indicators of these k-cycles. Itturnsoutthatc k 1 andc k nk+1 behavedifferentlythantherestofc k i ’s. Conditioningontheexistence of a k-cycle will result in two potential situations, depending on if the k-cycle is positioned in one of the end points, being the “first” or the “last” k-cycle or not. We will explain this through the bijection we defined in Chapter 3. Recall that there were 2 n1 b (2) -regular permutations, and compositions of n as well. We will condition on specific k-cycles’ existence, which in turn will be translated into existence of k-parts of the compositions. Then we count the number of compositions with that fixed k-part. Let’s first condition on the existence ofc k 1 orc k nk+1 . This conditioning leads us to compositions with ak-partinoneoftheendpoints, leavingnk remainingtobecomposed. Thereare 2 nk1 compositions of nk, therefore the probability that c k 1 (or, c k nk+1 ) is a k-cycle of the uniformly randomly selected 2S b(2) n , or equivalently, the probability that I k 1 (or, I k nk+1 ) occurs is as follows P (I k 1 occurs) =P (I k nk+1 occurs) = 2 nk1 2 n1 = 1 2 k : If we want one of the “mid” k-cycles to occur, then this would separate the remaining parts of the composition into two pieces, say of sizes a and nka, where a is a positive integer. Then, there are 2 a1 2 nka1 = 2 nk2 ways to compose these two parts, resulting in P (I k j occurs) = 2 nk2 2 n1 = 1 2 k+1 ; for j = 2; 3;:::;nk: 48 4.1.1 Dependence Structure of the Indicators Now let us go back to conditioning on the occurence of I k 1 . Firstly, it would be impossible for any other potential cycle that would use any number inc k 1 , namely 1; 2;:::;k to exist. Asc k 1 corresponds to ak-part, the resulting subspace will be identical to a composition of nk. Here, within this remaining nk part the probability for any k-part stays the same, except for the cycle that starts immediately after I k 1 . This k-cycle, c k k+1 used to be a “mid” k-cycle, with probability of happening 1=2 k+1 , but now its one of the endk-cycles of the remainingnk composition, bumping its probability of happening to 1=2 k . For j =k + 2;:::;nk + 1, I k j remain unaffected. Symmetrically, the occurence of the k-cycle c k nk+1 prevents k-cycles c k nk ;:::;c k n2k+2 would become impossible, and I k n2k+1 would change its probability of happening from 1=2 k+1 to 1=2 k . The remaining k-cycles would remain unaffected. A similar argument can be done forI k j , forj = 2;:::; (nk), where anyk-cycle that uses a value in I k j becomes impossible. Anyk-cycle with a value that is adjacent to a value inI k j changes its probability either from 1=2 k to 1=2 k+1 or vice versa. We now define a measure of distance between the cycles of a permutation that will help us capture the essence of the described dependency. Definition 4.1.1. Let c and d be any two different cycles of a permutation , where elements of c are c 1 ;c 2 ;:::;c k and elements of d are d 1 ;d 2 ;:::;d m . Then the cycles c and d are called p-separated if min i2f1;:::;kg j2f1;:::;mg jc i d j j>p and p is the largest number satisfying this inequality. Example 4.1.2. The two cycles (1542) and (789) are 1-separated, as 49 min i2f1;2;3;4g j2f1;2;3g jc i d j j =j5 7j> 1; but min i2f1;2;3;4g j2f1;2;3g jc i d j j =j5 7j 2: Similarly, (3) and (798) are 3-separated, as min i2f1g j2f1;2;3g jc i d j j =j3 7j> 3; but min i2f1g j2f1;2;3g jc i d j j =j3 7j 4: Combining these observations, we reach to the following independence structure. Proposition 4.1.3. [Theorem 2.12 in Ozel (2017)] Let c k 1 ;c k 2 ;:::;c k nk+1 be the potential k-cycles of a random b (2) -regular permutation. Then the indicators I k i and I k j are independent if and only if these two cycles are at least 1-separated, i.e.,jijj>k + 1 in terms of index of cycles. Proof. WLOG assuming i < j and using the same reasoning before, for the occurence of c k i = (i i + k i +k 1 i +k 2:::i + 1) not to affect occurence of c k j = (j j +k j +k 1 j +k 2:::j + 1), we must have the maximum valuei +k inc k i and the minimum valuej inc k j to be at least 1-separated. This happens when ji>k + 1. Thus,jijj>k + 1 implies the independence of the indicators for I k i and I k j (and vice versa). In order to visualize the dependence in Proposition 4.1.3, we provide the dependency graph for fixed point indicators and transposition indicators. If two nodes are connected in this graph, then the corresponding indicators are dependent. We consider b (2) -regular permutations for n = 6. The 50 possible fixed points are c 1 1 = (1); c 1 2 = (2); c 1 3 = (3); c 1 4 = (4); c 1 5 = (5); c 1 6 = (6) and the possible transpositions are c 2 1 = (12); c 2 2 = (23); c 2 3 = (34); c 2 4 = (45); c 2 5 = (56). I 1 1 ; (1) I 1 2 ; (2) I 1 3 ; (3) I 1 4 ; (4) I 1 5 ; (5) I 1 6 ; (6) Figure 4.1: Dependency Graph for Fixed Point Indicators In Figure 4.1 we can see that the indicators of fixed points follow 1-separation. For example the indicator for (2) is dependent to the indicators of (1) and (3), but not (4), (5) or (6). I 2 1 ; (12) I 2 2 ; (23) I 2 3 ; (34) I 2 4 ; (45) I 2 5 ; (56) Figure 4.2: Dependency Graph for Transposition Indicators 51 The picture gets more and more complicated for larger cycle sizes, but the principle is the same. Two cycle indicators are independent if they do not include the same number and they are at least one-separated. For b (r) -regular permutations for r 2, we can sense how Proposition 4.1.3 might be generalized. The restriction (i)ir + 1 creates “dependence blocks” for each k-cycle in the following sense. Conjecture 4.1.4. Let c k i and c k j be the two potential k-cycles of a random b (r) -regular permutation. ThentheindicatorsI k i andI k j areindependentifandonlyifthesetwocyclesareatleast (r1)-separated. This conjecture seems correct for many examples with small n, but a general proof is elusive at the moment. 4.1.2 m-dependence and Central Limit Theorems The above form of dependence is a slightly weaker version of the type of dependence referred to as “m-dependence”. Definition 4.1.5 (m-dependence). Letm be a non-negative integer,s andt positive integers witht<s andfX i g i2N be a sequence of random variables with the property that fX 1 ;X 2 ;:::;X t g? ?fX s ;X s+1 ;:::;X n g whenever st>m. ThenfX i g i2N is said to be m-dependent. Here the independence is presented in a joint distributional sense, rather than mutual independence as we considered. The proof of Proposition 4.1.3 can be generalized in a way that would satisfy m- dependence without much difficulty. As we will not require m-dependence and as it looks to become exponentially more difficult to prove m-dependence for b (r) -regular permutations, we will refrain from doing so. Central limit theorems for m-dependent random variables have been studied extensively, and have a vast literature. One of the earliest results on dependent central limit theorems is due to Bernstein 52 [7], in his paper “Sur l’extension du théorème limite du calcul des probabilités aux sommes de quantités dépendantes”. There are various results that followed Berstein’s steps. In 1948, Hoeffding and Robbins published “The Central Limit Theorem for Dependent Random Variables” [26], where they prove the central limit theorem form-dependent random variables satisfying certain regularity conditions, includ- ing having bounded third moments. Diananda (1954), in “The Central Limit Theorem form-Dependent Variables” [15] improves Hoeffding and Robbins by replacing finite third moment condition with weaker conditions, including having uniformly bounded variances. Similarly, Orey (1958), in “A Central Limit Theorem for m-Dependent Variables” [29] presents another result without the moment conditions, but equivalent conditions on sums of tail probabilities. An improved version of this sequence of results is due to A. Dvoretzky (1972). In “Asymptotic Normality for Sums of Dependent Random Variables” [16] regularity conditions in the preceding papers are relaxed significantly and the results are established for a more general definition of dependence. Of course, this is not an exhaustive list. 4.2 The Main Result We prove the central limit theorem for the number k-cycles of a uniformly randomly selected b (2) - regular permutation using one of the results in Chen and Shao’s paper, “Normal Approximation Under Local Dependence” [10]. They utilize Stein’s Method and establish distributional approximations for random variables that may be expressed as sums of locally dependent parts. They define this local dependence to an extent that is independent from the structure of the index set. Many theoretical and applicational advantages of Stein’s Method is beyond the scope of this thesis, but a great reference is the book by Chen, Goldstein and Shao “Normal Approximation by Stein’s Method” [11]. We also find Ross’s survey “Fundamentals of Stein’s Method” [32] quite enlightening. We initially used Theorem (3.6) in Ross [32], but Theorem (2.7) on [10] provides a better upper bound to the approximational distance. Before stating the main result we will provide some definitions and notation. 53 4.2.1 Probability Metrics Let X and Y be two random variables with the corresponding laws X and Y . A commonly used form of metrics in probability is d ( X ; Y ) = sup g2 Z g(x)d X (x) Z g(x)d Y (x) ; (4.2.1) where is a specified family of functions. We will also write d (X;Y ) instead of d ( X ; Y ), which is a slight abuse of notation that saves space in arguments. For this metric to be powerful in the distributional approximation sense, has to be a large enough family of functions. There are several natural choices for this family. 1. =f1[:2S]; S2 Borel(R)g, family of indicator functions for Borel measurable sets, yields the total variation metric, which is commonly used for approximation of discrete distributions. See the earlier mentioned paper by Arratia and Tavare (1992) [5] for more detail. 2. =f1[:2 (1;x]]; x2 Rg, family of right bounded infinite intervals, yields the Kolmogorov metric, which we denote asd K . Kolmogorov metric is one of the most commonly used probability metrics. One reason for this is that it gives the maximum distance between distribution functions and thus it is immediate that convergence in Kolmogorov distance implies weak convergence. Obtaining optimal bounds in this metric is often troublesome, but fortunately the result we use due to Chen and Shao [10] optimal. 3. =fg :R!R :jh(x)h(y)jjxyjg, family of real-valued Lipschitz functions with Lipschitz constant less than 1, or the family of real-valued contraction mappings, yields the Wasserstein distance or Kantorovich-Monge-Rubinstein metric, which we notate by d W . This metric is com- monly used for approximation of continuous distributions. See the earlier mentioned survey by Ross (2011) [32] for more detail. We originally used a result from Ross (2011), that provides the approximation in Wasserstein dis- tance, and using the following result, translates it into distributional distance via Kolmogorov metric. 54 Proposition 4.2.1 (Proposition 2.1 in Ross (2011)). Let Z be a random variable, whose Lebesgue density is bounded by a constant C, then for any random variable W, d K (W;Z) p 2Cd W (W;Z): Proof. Let x2R, consider the indicators function h x (t) = 1ft xg and its continuous version for a any > 0, h x; (t) = 8 > > > > > > > > < > > > > > > > > : 1; if t<x xt + 1; if xt<x + 0; if x +t: Then E[h x (W )]E[h x (Z)] =E[h x (W )]E[h x; (Z)] +E[h x; (Z)]E[h x (Z)] (4.2.2) E[h x; (W )]E[h x; (Z)] +C 2 (4.2.3) d W (W;Z) +C 2 ; (4.2.4) where (4.3.3) follows from the fact thath x hx; and the fact thath x andhx; only differ between x and x +, all the while the density of Z is bounded above by C. (4.3.4) follows from the fact that h x; (:) is a contraction mapping. Now, letting = p 2d W (W;Z)=C, we get E[h x (W )]E[h x (Z)] p 2Cd W (W;Z): Changing the roles of W and Z and taking the supremum over x2R we get the desired result. 55 4.2.2 Two Graph Dependency CLTs and the Result We start this section by stating the two versions of the graph dependency results given in [32] and [10], then compare the error bounds that we obtain. The following is the first version. Theorem 4.2.2. [Theorem 3.6 in Ross (2011)] Let X 1 ;X 2 ;:::;X n be random variables such that E[X 4 i ]<1,E[X i ] = 0, 2 =Var( P i X i ), anddefineW = P i X i =. Letthe collection (X 1 ;X 2 ;:::;X n ) have dependency neighborhoods N i , i = 1; 2;:::;n, and also define D := max 1in jN i j. Then for Z a standard Normal random variable, d W (W;Z) D 2 3 n X i=1 EjX i j 3 + p 28D 3=2 p 2 v u u t n X i=1 E[X 4 i ]: We recall some facts of the k-cycle indicators, I k i , i = 1; 2;:::;nk + 1 we observed back in Section 4.1. Here, N i is defined to be the set of random variables X j , which are dependent to X i . By Proposi- tion 4.1.3,jN i j may be k;k + 1;:::; 2k, therefore D = max 1in jN i j = 2k: Also, recall the probabilities associated to the indicator occurences. P (I k i = 1) = 8 > > > < > > > : 1=2 k ; for k = 1;nk + 1 1=2 k+1 ; for k = 2; 3;:::;nk Now we define the shifted version indicators. X k i = 8 > > > < > > > : I k i 1=2 k i = 1;nk + 1 I k i 1=2 k+1 i = 2;:::;nk (4.2.5) 56 so that E[X k i ] = 0. Recall, from Chapter 3, that the expected number of k-cycles for a uniformly randomly selected b (2) -regular permutation,E[C n;k ] is n;k =E[C n;k ] = nk + 3 2 k+1 : Also, the variance for C n;k is 2 n;k =Var(C n;k ) = (2 k+1 2k + 3)n + 3k(k 4) + (3k)2 k+1 + 5 4 k+1 : Variance of the sum of these shifted indicators is still 2 n;k as the sum of these shifts still does not change the variance, and n;k is this total shift amount. What we need to calculate are EjX k i j 3 and E[(X k i ) 4 ] for each i. By the definition of X k i , these values are different for i = 1;nk + 1 versus i = 2;:::;nk. For i = 1;nk + 1, jX k i j = 8 > > > < > > > : 1=2 k with probability1 1 2 k 1 1=2 k with probability 1 2 k and for i = 2;:::;nk, jX k i j = 8 > > > < > > > : 1=2 k+1 with probability1 1 2 k+1 1 1=2 k+1 with probability 1 2 k+1 : Calculating their third moments, for i = 1;nk + 1, EjX k i j 3 = 1 2 3k 1 1 2 k + 1 1 2 k 3 1 2 k = 2 3k 3:2 2k + 2 k+2 2 2 4k ; 57 and for i = 2;:::;nk, EjX k i j 3 = 1 2 3k+3 1 1 2 k+1 + 1 1 2 k+1 3 1 2 k+1 = 2 3(k+1) 3:2 2(k+1) + 2 k+3 2 2 4(k+1) : Summing these over i, we get a n;k := nk+1 X i=1 EjX k i j 3 = 2 2 3k 3:2 2k + 2 k+2 2 2 4k + (nk 1) 2 3(k+1) 3:2 2(k+1) + 2 k+3 2 2 4(k+1) Now we calculate the fourth moments. For i = 1;nk + 1, E[(X k i ) 4 ] = 1 2 k 4 1 1 2 k + 1 1 2 k 4 1 2 k = (2 2k 32 k + 3)(2 k 1) 2 4k ; and for i = 2;:::;nk, E[(X k i ) 4 ] = (2 2(k+1) 32 k+1 + 3)(2 k+1 1) 2 4(k+1) : Summing these over i, we get 58 b n;k := nk+1 X i=1 E[(X k i ) 4 ] = 2 (2 2k 3:2 k + 3)(2 k 1) 2 4k + (nk 1) (2 2(k+1) 3:2 k+1 + 3)(2 k+1 1) 2 4(k+1) : Now we combine all these ingredients within Theorem 4.2.2. Theorem 4.2.3. [Theorem 2.14 in Ozel (2017)] Let be a uniformly randomly selected b 2 -regular permutation,X k i be the shifted indicators ofk-cycles of and 2 = 2 n;k . Further, defineW = 1 P i X k i . Then for ZN(0; 1), d W (W;Z) 4k 2 3 a n;k + p 28(2k) 3=2 p 2 p b n;k ; where 2 = (n), a n;k = (n) and b n;k = (n). Therefore the upper bound is of (n 1=2 ). Asymptotic Normality of C n;k follows immediately from Theorem 4.2.3. But the order of the upper bound is not optimal. By Proposition 4.2.1, the upper bound for the Kolmogorov distance is only of order ( 1 n 1=4 ). Using the graph dependency result in Chen and Shao (2004) gives a significantly better upper bound. The following is the second version. Theorem 4.2.4. [Theorem 2.7 in Chen and Shao (2004)] LetfX i ;i2Vg be random variables indexed by the vertices of a dependency graph G, with the maximal degree of G being D. Put W = P i2V X i . Assume that E[W 2 ] = 1, E[X i ] = 0. If there exists some > 0 EjX i j p p for all i2 V and all 2<p 3, then for ZN(0; 1), d K (W;Z) 75D 5(p1) jVj p (4.2.6) and for z2R, jP (Wz) (z)jC (1 +jzj) p D 5p jVj p : (4.2.7) 59 The set up similar to before. We let X k i to be as in Equation 4.2.5. But here we will scale X k i by to satisfy the condition E[W 2 ] = 1. Let Y k i = X k i =, the shifted and scaled version of the k-cycle indicators. Letting W = P nk+1 i=1 Y i , we already haveE[W ] = 0, as X k i were shifted the right amount, and thus so are Y k i . Also, with the appropriate scaling we haveE[W 2 ] = 1. Now as X k i = 8 > > > < > > > : I k i 1=2 k i = 1;nk + 1 I k i 1=2 k+1 i = 2;:::;nk ; we have jX k i j< 1 1 2 k : Thus, jY k i j = jX k i j < 1 1=2 k ; so we will set to be this upper bound, i.e., = 1 1=2 k : In order to minimize the upper bound, we set p = 3 and proceed to the optimal result. Theorem 4.2.5 (The Main Result). Let be a uniformly randomly selected b (2) -regular permutation, Y k i be the shifted and scaled indicators of k-cycles of and 2 = 2 n;k . Further, define W = 1 P i Y k i . Then for ZN(0; 1), d K (W;Z) 75 (2k) 10 (nk + 1) 1 1=2 k 3 : 60 We can reorganize this upper bound as d K (W;Z)f(k) nk + 1 3 n;k ; where f(k) = 75 (2k) 10 (1 1=2 k ) 3 . As 3 n;k = (n 3=2 ), the upper bound is essentially of order (n 1=2 ). This is a significant improvement over the bound in Theorem 4.2.3, which was (n 1=4 ). 61 Chapter 5 Conclusion and Future Work Our main result, Theorem 4.2.5 implies the following corollary, which is also proved for the first time. Corollary 5.0.1. Let be a uniformly randomly selected composition of n, where n is a large integer. Then the number of k-parts of , say M k , M k N n;k ; 2 n;k : Here, the distributional approximation is as in Theorem 4.2.5. As mentioned before we would like to observe the asymptotic behavior of k-cycles of b (r) -regular permutations for an arbitrary positive integer r. For r = 2 we have the corresponding results, but what about r > 2? Intuitively speaking we would expect quite similar results, asymptotic Normality, expected value and variance as linear functions of n, etc. This is because of the similar, finitistic restriction structures b (r) -regular permutations propose. Recall that in Chapter 4 we found out that k-cycle indicators were identical up to being in the end-most positions. Indicators satisfied a symmetry of the form A 1 A 2 A 2 :::A 2 A 2 A 1 : 62 It turns out that this structure of indicators is generalized alongside r. For r = 3 we observed that the indicators have the following form A 1 A 2 A 3 A 3 :::A 3 A 3 A 2 A 1 ; and for r = 4, A 1 A 2 A 3 A 4 A 4 :::A 4 A 4 A 3 A 2 A 1 ; so on and so forth. Here A i differ due to occurence probabilities. As mentioned in Section 4.1.1 we also have a conjecture for the dependence structure of the indicators that holds true for trials with small n. Conjecture 5.0.2. Let c k i and c k j be the two potential k-cycles of a random b (r) -regular permutation. ThentheindicatorsI k i andI k j areindependentifandonlyifthesetwocyclesareatleast (r1)-separated. In order to further consolidate our intuitive predictions we conducted simulations and plotted the histogram of number ofk-cycles. In the next section we will describe the algorithm we used to produce a random b (r) -regular permutation. 5.1 Simulation Results Given any one-sided restriction vector b, Diaconis et al (2001) provides a method that generates a uniformly random b-regular permutation. Proposition 5.1.1 (Lemma 3.2 in Diaconis et al (2001) [13]). Letb be a restriction vector defined as in Chapter 2, S b n be the set of permutations in accordance with this restriction vector. Then the following algorithm results in a uniform choice from S b n . Begin with the whole set of positions,f1; 2;:::;ng. Map position n uniformly randomly to an allowed label, from the set L n =fj :jb n g. 63 Let L n1 =fj :jb n1 gf(n)g. Map position n 1 uniformly randomly to an allowed label in L n1 . Let L n2 =fj : j b n2 gf(n);(n 1)g. Map position n 2 uniformly randomly to an allowed label in L n2 . . . . Let L 1 =fj : j b 1 gf(n);(n 1);:::;(2)g. Map position 1 uniformly randomly to an allowed position in L 1 , in which case there is a unique label left. Proof. Recall that the entries of a restriction vector were always less than or equal to the corresponding index, i.e., b i i. The choice of the allowed and unused labels from the corresponding options L i is always possible as this set is never empty due to the fact that b i i WLOG. Therefore the algorithm produces an element of S b n without getting stuck. Conversely, any specific b-regular permutation, 2S b n can be constructed by a sequence of specific choices. Every step in the algorithm is deterministic except the uniform choice in each step, thus every 2S b n is equally likely. The algorithm in Proposition 5.1.1 allows us to generate a simple random sample of b (r) -regular permutations, count their cycles and plot the histogram. We achieved this using Matlab, for which the code is included in the appendix. We will now present the simulation results for the number of fixed points, when r = 3, r = 5 and r = 100. We also plot the target Normal distribution for reference. In each case the Normal distribution seems to appear, which motivates us to further the investigation. 64 Figure 5.1: Histogram of fixed points for b (3) -regular permutations on 1000 letters Figure 5.2: Histogram of fixed points for b (5) -regular permutations on 1000 letters 65 Figure 5.3: Histogram of fixed points for b (100) -regular permutations on 10000 letters 5.2 Colored Compositions and the Random Walk Structures As we have seen in Chapter 3 and Chapter 4, the bijection between S b (2) n and compositions ofn was crucial in establishing our main result. So it would be a good idea to identify combinatorial objects with similar utility for b (r) -regular permutations with r> 2. Recalling that jS b (r) n j =r!r nr ; we checked OEIS (The Online Encyclopedia of Integer Sequences) for the sequence corresponding to r = 3,f3! 3 n3 g n0 . The entry by Adams-Watters identifies that this sequence corresponds to the number of compositions of n into parts of two kinds [2]. We will consider these two “kinds” in terms of colors, i.e., each part of the composition is either “black” or “white”. This definition is different than the classical problem studied by Agarwal (2000) “n-Color Compositions” [3], where each k-part might assume k different colors. We consider a fixed number of colors for parts of any size. Abrate, Barbero, Cerruti and Murru relatively recently (2014) 66 worked on the most general form of colored compositions by introducing any positive integer sequence to be a coloration, whereith element of the sequence determines the number of colors eachi-part might take on. So Agarwal’s n-color compositions would be expressed via the colorization C A = (1; 2; 3; 4;:::); whereas our two-colored compositions above would be described via the colorization C 2 = (2; 2; 2; 2;:::): In their paper “Colored compositions, Invert operator and elegant compositions with the ‘black tie’ ” [1], Abrate et al present a nice method of checking if any given sequence of positive integers correspond to a colorization of compositions of n. We will use this to identify the colored compositions we might work with. First, we define the “Invert operator”. Definition 5.2.1. The Invert operator I transforms a sequence a =fa n g n0 into a sequence b = fb n g n0 as follows: I(a) =b; 1 X n=0 b n t n = P 1 n=0 a n t n 1t P 1 n=0 a n t n : Theorem 5.2.2 (Theorem 3 in Abrate et al (2014) [1]). For every sequencefa n g n1 with generating function a(t), a n is the number of colored compositions of n, for any n 1, with coloration X if and only if I 1 (a) is a sequence of nonnegative integers and in this case X = I 1 (a), where I 1 (a) has generating function a(t) 1+ta(t) . We now have the tools to check if any sequence corresponds to a colorization. Here are two examples we know to be true. 67 Example 5.2.3. Consider the sequence 2 n1 forb (2) -regular permutations. With one unit of correction factor to the index, we obtain a n = 2 n and the generating function, a(t) = X n0 2 n t n = 1 1 2t : Then the Invert operator applied to a(t) becomes I 1 (a) = 1 12t 1 +t 1 12t = 1 1 2t +t = 1 1t = X n0 1t n : Therefore the sequencef2 n1 g n1 corresponds to the colorization of compositions of n with only 1 color, i.e., compositions of n themselves. Example 5.2.4. If we are to apply this process to Fibonacci sequence without the first entry, I 1 (1; 2; 3; 5; 8;:::) = (1; 1; 0; 0;:::); we obtain compositions of one color, only with part sizes of 1 and 2. This is in accordance with Proposition 2.3.2, where we stated Fibonacci permutations are in one-to-one correspondence with com- positions that only have parts 1 and 2. If we are to do the same for regular Fibonacci sequence, we obtain I 1 (1; 1; 2; 3; 5; 8;:::) = (1; 0; 1; 0; 1; 0;:::); the compositions that only use odd part sizes. Now we check out b (3) -regular permutations. Example 5.2.5. Consider 3! 3 n3 = 2 3 n2 . After the appropriate shift to the index, we let a n = 2 3 n the generating function, a(t) = X n0 2 3 n t n = 2 1 1 3t : 68 Applying the Invert operator a(t), we get I 1 (a) = 2 13t 1 + 2t 1 13t = 1 1 2 (1 3t) +t = 1 1=2t=2 = 2 1t = X n0 2t n : Therefore the sequencef3! 3 n3 g n2 corresponds to the colorization of compositions of n 1 with 2 colors, i.e., what we refer to as two-colored compositions. There is a bijection from b (2) -regular permutations to one-colored compositions of n and there is a bijectionfromb (3) -regularpermutationstotwo-coloredcompositionsn1. Soitisnaturaltoexpectthat thereisabijectionfromb (r) -regularpermutationsto (r1)-coloredcompositionsnr+2. Unfortunately this turns out to be incorrect. For r > 3 this relation seems not to be hold. Actually, the number of (r 1)-colored compositions systematically undercount the b (r) -regular permutations. One way to fix this problem is via introducing an extra layer of hierarchy (or ordering) between all the color except one, which provides the exact coefficient of repetition (r 1)!. Another approach would be to identify other combinatorial objects that would better explain the structure of b (r) -regular permutations. 5.2.1 Random Transposition Walk on Restricted Permutations One way to identify the bijection between the b (3) -regular permutations and 2-colored compositions might be through the structural equivalence between the random walks over these two sets of objects. To express this point clearly, we go back to b (2) -regular permutations and compositions, and have a general look at the random transposition walk over general b-regular permutations. A most influential work on the random transposition walk on S n is the 1981 paper “Generating a Random Permutation with Random Transpositions” by Diaconis and Shahshahani [14]. The walk is described as follows. Start from the identity permutation, at each step imagine choosing one of the numbersf1; 2;:::;ng uniformly at random with left hand and, independently, one with right hand. If we obtained two different numbersi andj, then we apply the transposition (ij) to the current permutation. If i =j, then the walk stays at the current permutation. Diaconis and Shahshahani provides an upper bound to the total variation distance between this walk and the uniform distribution over S n , using 69 tools from Representation Theory. They conclude that doing this walk about 1 2 n log(n) times results in a uniformly random selection from S n . Hanlon (1996) [22] and Blumberg (2011, 2012) [9] [8] extend this random walk to permutations with positional restrictions. In the most general sense, we do the above random transposition walk, but any time the resulting permutation does not adhere to the restriction matrix M, the walk stays. If the resulting permutation after the application of the random transposition is in accordance with M, we proceed. Hanlon proves that for one-sided restrictions this random walk has a regular graph structure. Proposition 5.2.6 (Proposition 2.1(d) in Hanlon 1996 [22]). Let b a restriction vector for S n . Then the graph corresponding to the random transposition walk on S B n is a regular graph of degree n X i=1 (ib i ) = n 2 n X i=1 (b i 1): For b (r) -regular permutations this gives a regular graph of degree (r 1) (n r 2 ): (5.2.1) For example, considering the r = 2 case, we get (2 1) (n 2=2) =n 1. This means, at whatever b (2) -regular permutation we might be, there aren1 otherb (2) -regular permutations we might move to, through an appropriate transposition. Now we will define a random walk over the set of compositions to compare to the b (2) -regular transposition walk. Let = 1 + 2 +::: + m be any composition ofn. Then we define two possible random walk steps as (i) Compose any two adjacent parts together: for any i = 1; 2;:::;m 1, create a new part i by the sum i = i + i+1 . (ii) Take any part into two smaller parts: for any i = 1; 2;:::;m, create two new parts, i and i+1 by the decomposition i = i + i+1 . 70 Now we will explain why this random walk has a regular graph of degree n 1. Imagine we are at the composition = 1 + 2 +::: + m . Then there are m 1 possible ways to take step (i), as there arem 1 potential sums between i . On the other hand, for each i , there are i 1 ways to separate the part into two pieces, as deciding the size of the first part determines the size of the second part, and further, there are i 1 possibilities for the size of the first part, namely, 1; 2;:::; i 1. So, overall, we have (m 1) + m X i=1 ( i 1) potential steps to take. Recalling that P m i=1 i =n as is a composition of n, we get (m 1) + m X i=1 i ! m =n 1 potential steps, which implies the n 1 regularity of this random walk graph. Furthermore, due to the cycle structure of b (2) -regular permutations it is easy to see that our bijection translates any two adjacent compositions to two adjacent permutations in their corresponding random walks. We will demonstrate this with an example. Example 5.2.7. Consider 2S b (2) 5 , in cycle notation = (1432)(5). Recall that this is mapped to the composition 4 + 1. There are 4 possible transpositions we can apply to and still have a b (2) -regular permutation as the result, namely, (12); (13); (14) and (15). In the first three cases, we obtain (1) = (1)(243)(5), (2) = (12)(34)(5) and (3) = (132)(4)(5), with the corresponding compositions of 1 + 3 + 1, 2 + 2 + 1 and 3 + 1 + 1, which are exactly 4 + 1 with a potential (ii)-step. The transposition (15) would result in (4) = (15432), i.e., the composition we obtain by the only possible (i)-step, 5. We believe that a similar structural equivalence is preserved for b (3) -regular permutations and two- colored compositions of n 1. Random transposition walk over S b (3) n is the same as any S M n , we allow 71 a transposition step if the resulting permutations is still b (3) -regular, otherwise we stay. Now we define the random walk over two-colored compositions. Let = 1 + 2 +::: + m be any two-colored composition of n 1, where each i might have two colors, say black and white. We will denote a black part by an underline, e.g., a black part of size five is 5 versus a white part of size five 5. In the random walk we define, there are three possible steps to take. (i) Change the color of any part: for any i = 1; 2;:::;m, change i to i or i to i . (ii) Similar to one-colored compositions, compose any two adjacent parts together. But in the case of two-colored compositions we have a new subtle detail. WLOG, black parts behave like even numbers, whereas white parts behave like odd numbers in the sense that sum of two odd or two even parts is even, whereas the sum of an odd and an even part is odd. (iii) Similar to one-colored compositions, take any part into two smaller parts. The even and odd rule above is still in play. An odd part might be separated into an odd and an even part, where as an even part would result in two odd or two even parts. By Equation 5.2.1 we know that for the S b (3) n random walk, the graph should be regular of degree (31) (n 3 2 = 2n3). For theabove two-coloredcompositionsn1 walk, sayweare atthe composition = 1 + 2 +::: + m , with m X i=1 i =n 1: There are m (i)-steps as for each part we can switch to the other color. Further, we have m 1 (ii)-steps, as whatever the colors are there are m 1 possible additions. Lastly, for each part i , black or white, we have 2( i 1) possible decompositions into two parts, as if i is white, we have i 1 ways to decompose in the one-colored composition sense, but for each decomposition the first part might be white and the second black, or the other way around. If i is even, we have a similar repetition caused by parts being both black or both white. Combining (i)-steps, (ii)-steps and (iii)-steps, 72 m + (m 1) + m X i=1 2( i 1) = 2m 1 + 2 m X i=1 i ! 2m = (2m 1) 2m + 2(n 1) = 2n 3 as desired. There are further clues towards the correct bijection. It turns out that the distribution of total number of cycles for b (3) -regular permutations is equivalent to the distribution of number of (WLOG) black parts +1 for two-colored compositions. Example 5.2.8. The elements of S b (3) 4 are the following 18 permutations. Permutation Total Number of Cycles (1)(2)(3)(4) 4 (12)(3)(4) 3 (1)(23)(4) 3 (1)(2)(34) 3 (13)(2)(4) 3 (1)(24)(3) 3 (12)(34) 2 (13)(24) 2 (1)(234) 2 (1)(243) 2 (143)(2) 2 (142)(3) 2 (123)(4) 2 (132)(4) 2 (1432) 1 (1423) 1 (1243) 1 (1342) 1 Table 5.1: b (3) -regular permutations off1; 2; 3; 4g Here, distribution for the total number of cycles is (4; 8; 5; 1), i.e., there are 4 permutations with 1 cycle, 8 permutations with a total of 2 cycles, 5 permutations with a total of 3 cycles and lastly, 1 permutation, the identity, with a total of 4 cycles. Now we will compare this distribution to the number of black parts in two-colored compositions. 73 The two-colored compositions of 4 1 = 3 are as follows. Two-colored Composition Number of Black Parts + 1 3 1 3 2 2 + 1 1 2 + 1 2 2 + 1 2 2 + 1 3 1+2 1 1 + 2 2 1 + 2 2 1 + 2 3 1 + 1 + 1 1 1 + 1 + 1 2 1 + 1 + 1 2 1 + 1 + 1 2 1 + 1 + 1 3 1 + 1 + 1 3 1 + 1 + 1 1 1 + 1 + 1 4 Table 5.2: Two-colored compositions of 3 Counting the numbers on the second column, we observe that we have the same list of numbers as the second column of Table 5.1, i.e, (#1 0 s; #2 0 s; #3 0 s; #4 0 s) = (4; 8; 5; 1). 5.3 Concluding Remarks With the help of random walk structure and the association between the number of black parts and the number cycles we aim to identify the bijection. But at this stage the problem poses an extra layer of challenge. In b (2) -regular permutations there is a unique structure of cycles, in the sense that compositions, as they are, are enough to capture the cycle structures of these permutations. But b (3) - regular permutations are not enough to capture the multiplicities of cycle structures. For instance, in the above Example 5.2.8 we saw that there are 4 permutations with only a 4-cycle. So, on top of the bijection we need a rule to switch from two-colored compositions to cycle types of b (3) -regular permutations. 74 There might be many further difficulties we are now aware of in the study of permutations with restricted positions, but as we have seen throughout this dissertation, these difficulties are not without reward. Weplantocontinueourstudyfurtherintob (r) -regularpermutationswithr> 2, tobefollowedbythe casewherer =r(n)arefunctionsofnwithvariousbehavior. Itwillbeinterestingtoobservethechanges in the cycle structure depending on r. Another idea is to do similar analyses for other permutation statistics, such as the number of inversions and the length of the longest increasing subsequences to be compared to their respective classical results. Lastly, as mentioned in the previous section, the random walk over the permutations with specific restrictions are studied in literature. One of the most extensive works in this direction is presented in Olena Blumberg’s doctorate thesis [9]. She mainly studied interval restrictions, where in each row of the restriction matrix, entries of 1’s occur consecutively, within an interval. Two-sided and one-sided restrictions are special cases of interval restrictions. She provides the mixing time of this random walk forgeneralboundedintervalrestrictionsaswellassomecasesofunboundedone-sidedrestrictions. These results, however, does not cover the random walks forb (r) -regular permutations. Thus, the study of the random walks over a more general subset of one-sided restrictions is another viable research direction. 75 Reference List [1] Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru. Colored compositions, invert operator and elegant compositions with the “black tie”. Discrete Mathematics, 335:1 – 7, 2014. [2] Franklin T. Adams-Watters. Number of compositions of n into parts of two kinds. https://oeis.org/A025192, August 2006. [3] A. K. Agarwal. N-colour compositions. Indian Journal of Pure and Applied Mathematics, 31(11):1421 – 1427, 2000. [4] R. Arratia, A.D. Barbour, and S. Tavaré. Logarithmic Combinatorial Structures: A Probabilistic Approach. EMS monographs in mathematics. European Mathematical Society, 2003. [5] Richard Arratia and Simon Tavare. The cycle structure of random permutations. The Annals of Probability, 20(3):1567–1591, 1992. [6] Vladimir Baltic. On the number of certain types of strongly restricted permutations. Applicable Analysis and Discrete Mathematics, 4(1):119–135, 2010. [7] S. Bernstein. Sur l’extension du theoreme limite du calcul des probabilites aux sommes de quantites despendantes. Mathematische Annalen, 97:1–59, 1927. [8] O. Blumberg. Cutoff for the Transposition Walk on Permutations with One-Sided Restrictions. ArXiv e-prints, February 2012. [9] O. Bormashenko. Permutations with Interval Restrictions. PhD thesis, Stanford University, De- partment of Mathematics, 2011. [10] Louis H. Y. Chen and Qi-Man Shao. Normal approximation under local dependence. The Annals of Probability, 32(3):1985–2028, July 2004. [11] Louis H.Y. Chen, Larry Goldstein, and Qi-Man Shao. Normal Approximation by Stein’s Method. Springer-Verlag Berlin Heidelberg, 2011. [12] P.R. de Montmort, S. Leclerc, and P.L. Sueur. Essay d’analyse sur les jeux de hasard [par P. Rémond de Montmort]. Jacques Quillau, 1708. [13] Persi Diaconis, Ronald Graham, and Susan P. Holmes. Statistical Problems Involving Permutations with Restricted Positions, volume Volume 36 of Lecture Notes–Monograph Series, pages 195–222. Institute of Mathematical Statistics, Beachwood, OH, 2001. [14] Persi Diaconis and Mehrdad Shahshahani. Generating a random permutation with random trans- positions. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 57(2):159–179, June 1981. [15] P. H. Diananda. The central limit theorem for m-dependent variables. Mathematical Proceedings of the Cambridge Philosophical Society, 51(1):92–95, 1955. 76 [16] Aryeh Dvoretzky. Asymptotic normality for sums of dependent random variables. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Volume 2: Probability Theory, pages 513–535, Berkeley, Calif., 1972. University of California Press. [17] Jack Edmonds. Paths, trees, and flowers. Canad. J. Math., 17:449–467, 1965. [18] B. Efron and V. Petrosian. A simple test of independence for truncated data with applications to redshift surveys. apj, 399:345–352, November 1992. [19] Jay R. Goldman, J. T. Joichi, and Dennis E. White. Rook theory. i.: Rook equivalence of ferrers boards. Proceedings of the American Mathematical Society, 52(1):485–492, 1975. [20] V. L. Goncharov. Sur la distribution des cycles dans les permutations. C. R. (Doklady) Acad. Sci. URSS (N.S.), 35:267–269, 1942. [21] V. L. Goncharov. Some facts from combinatorics. Izvestia Akademii Nauk SSSR, 8 : 3-48, 1944. [22] Phil Hanlon. A random walk on the rook placements on a ferrers board. The Electronic Journal of Combinatorics [electronic only], 3(2), 1996. [23] S. Heubach and T. Mansour. Combinatorics of Compositions and Words. Discrete Mathematics and Its Applications. CRC Press, 2009. [24] P. Hitczenko and G. Stengle. Expected number of distinct part sizes in a random integer compo- sition. Combinatorics, Probability and Computing, 9(6):519–527, 2000. [25] Pawel Hitczenko and Carla D. Savage. On the multiplicity of parts in a random composition of a large integer. SIAM Journal on Discrete Mathematics, 18(2):418–18, 2004. [26] WassilyHoeffdingandHerbertRobbins. Thecentrallimittheoremfordependentrandomvariables. 15(3):773–780, 09 1948. [27] Mark Jerrum, Alistair Sinclair, and Eric Vigoda. A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries. J. ACM, 51(4):671–697, July 2004. [28] T Klove. Generating functions for the number of permutations with limited displacement. 16:1–11, 08 2009. [29] StevenOrey. Acentrallimittheoremform-dependentrandomvariables. Duke Math. J., 25(4):543– 546, 12 1958. [30] E. Ozel. The Number of k-Cycles In a Family of Restricted Permutations. ArXiv e-prints (arXiv:1710.07885), October 2017. [31] J. Riordan. Introduction to Combinatorial Analysis. Dover Books on Mathematics. Dover Publi- cations, 2002. [32] Nathan Ross. Fundamentals of stein’s method. Probab. Surveys, 8:210–293, 2011. [33] H. J. Ryser. Combinatorial mathematics. Carus mathematical monographs. Mathematical Associ- ation of America; distributed by Wiley [New York], 1963. [34] Richard P. Stanley. Enumerative Combinatorics: Volume 1. Cambridge University Press, New York, NY, USA, 2nd edition, 2011. [35] L.G.Valiant. Thecomplexityofcomputingthepermanent. Theoretical Computer Science,8(2):189 – 201, 1979. 77 Appendix A Matlab Code for Simulation 1 function b = rones(n, r) 2 %constructs the restriction vector of size n with r ones and increasing by 3 %one at each component 4 5 b= ones(n,1) ; 6 c= 1; 7 8 for i= r+1:n 9 b( i )= b( i )+c; 10 c= c+1; 11 end 12 13 b= b’; 1 function simulation 2 %returns the histogram of the number of fixed points of a randomly selected 3 %restricted permutation , where the restriction vector is b 4 close all ; 5 clc ; 6 tic ; 7 n= 1000; 8 b= rones(n,100) ; 9 trials= 100; 10 id= 1:n; 11 record= zeros( trials ,1) ; 12 list= 1:n; 13 14 for i= 1: trials 15 auxlist= list ; 16 perm= zeros(n,1) ; 17 18 for k= n:1:1 19 bauxlist= auxlist (auxlist>=b(k)) ;%set of allowed labels at this iteration 20 currentlabel= datasample(bauxlist ,1) ;%choosing pi(k) uniformly randomly 21 perm(k)= currentlabel ; 22 %fprintf(’%d got deleted ! ’ , currentlabel ) ; 78 23 auxlist ( auxlist==currentlabel )= []; 24 end 25 26 record( i )= sum(perm’==id) ; 27 end 28 29 figure 30 histfit (record ,[] , ’normal ’) ; 31 title ( ’r = 100 ’) 32 toc ; 79
Abstract (if available)
Abstract
In this dissertation we study cycles of permutations with restricted positions. Permutations with restricted positions have strong connections to many other fields of mathematics, including Complexity Theory and Graph Theory, as well as applications in Coding Theory and Mathematical Statistics. Our objective is to identify the relationship between the positional restrictions and the cycle structures. Specifically, we aim to identify the asymptotic distribution of the random variable Cₙ,ₖ, the number of k-cycles of a uniformly randomly selected permutation adhering to the given positional restriction. For the general one-sided restrictions we provide a method to calculate the moments of the Cₙ,₁, the number of fixed points. Considering a certain family of one-sided restrictions we prove the central limit theorems for Cₙ,ₖ, establishing the Normal asymptotic distribution with corresponding values of mean and variance. To this end, we utilize an application of Stein’s Method for locally dependent random variables. Further, our investigation leads to the discovery of bijections that relates permutations with restricted positions to compositions of positive integers and their generalizations. These connections allow the translation of results in one set of objects to the other, providing another important motivation to further this inquiry.
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Ozel, Enes
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Cycle structures of permutations with restricted positions
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Doctor of Philosophy
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Applied Mathematics
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04/09/2018
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cycles of a permutation
distributional approximation
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permutations with restricted positions
Stein's method