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ON TOP TO RANDOM SHUFFLES, NO FEEDBACK CARD GUESSING, AND
FIXED POINTS OF PERMUTATIONS
by
Lerna Pehlivan
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Ful llment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(MATHEMATICS)
December 2009
Copyright 2009 Lerna Pehlivan
Object Description
| Title | On top to random shuffles, no feedback card guessing, and fixed points of permutations |
| Author | Pehlivan, Lerna |
| Author email | lepehli@gmail.com; pehlivan@usc.edu |
| Degree | Doctor of Philosophy |
| Document type | Dissertation |
| Degree program | Mathematics |
| School | College of Letters, Arts and Sciences |
| Date defended/completed | 2009-07-15 |
| Date submitted | 2009 |
| Restricted until | Unrestricted |
| Date published | 2009-09-10 |
| Advisor (committee chair) | Fulman, Jason |
| Advisor (committee member) |
Baxendale, Peter Kempe, David |
| Abstract | We study two problems related to top to random shuffling. First, we show that for an O(n\ln(n)) top to random shuffled deck, there exists a best no feedback card guessing strategy, i.e., a strategy maximizing the expected number of correct guesses without revealing information to the guesser. Moreover, using the best no feedback guessing strategy, the difference between the expected number of correct guesses for a deck that is O(n\ln(n)) top to random shuffled and a deck that is uniformly distributed can be made arbitrarily small.; Second, we study the expected value and the variance of the number of fixed points of a permutation that is obtained after an arbitrary number of top to random shuffles and provide closed formulas for both. Two different methods of proofs are provided, one combinatorial and another one using the distribution of the eigenvalues among the irreducible representations of S_n. Since the number of fixed points is a feature of a permutation, it is natural to expect that with O(n\ln(n)) shuffles it will converge to Poisson(1). However, we show that O(cn) top to random shuffles (contrary to riffle shuffles) are not enough to achieve convergence to a Poisson(1). |
| Keyword | Markov chains; top to random shuffles; representation theory of the symmetric groups |
| Language | English |
| Part of collection | University of Southern California dissertations and theses |
| Publisher (of the original version) | University of Southern California |
| Place of publication (of the original version) | Los Angeles, California |
| Publisher (of the digital version) | University of Southern California. Libraries |
| Provenance | Electronically uploaded by the author |
| Type | texts |
| Legacy record ID | usctheses-m2596 |
| Rights | Pehlivan, Lerna |
| Repository name | Libraries, University of Southern California |
| Repository address | Los Angeles, California |
| Repository email | http://www.usc.edu/isd/libraries/services/ask_a_librarian/email/ |
| Filename | etd-Pehlivan-3239 |
| Archival file | uscthesesreloadpub_Volume40/etd-Pehlivan-3239.pdf |
Description
| Title | Page 1 |
| Full text | ON TOP TO RANDOM SHUFFLES, NO FEEDBACK CARD GUESSING, AND FIXED POINTS OF PERMUTATIONS by Lerna Pehlivan A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Ful llment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (MATHEMATICS) December 2009 Copyright 2009 Lerna Pehlivan |
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