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Seminar Series: Steven N. Evans

Steven Evans
November 12, 2020
All Day
Virtual

Title

Unseparated pairs and fixed points in random permutations

Meeting Link

Speaker

Steven N. Evans, University of California-Berkeley, Department of Statistics

Abstract

The result of shuffling a deck of $n$ cards is to re-order the cards so that the card originally in position $k$ is now in position $\Pi(k)$ for $1 \le i \le n$.  Experiments by Persi Diaconis and his collaborators looked at a particular shuffling mechanism called smoosh shuffling and asked how close the permutation $\Pi$ produced by smoosh shuffling was to being uniform (that is, to all $n!$ permutations being equally likely).  A question that arose out of this investigation is the following: What is the distribution of the random set of indices $\{1 \le i \le n-1: \Pi(k+1) = \Pi(k) +1\}$ when $\Pi$ is uniform?  That is, what is the distribution of the collection of successive pairs of cards that don't get separated by the shuffling procedure?  We'll discuss three different proofs of the remarkable fact that this random set of indices has the same distribution as $\{1 \le i \le n-1: \Pi(k) = k\}$, the set of cards in the first $n-1$ that don't change position.  This is joint work with Persi Diaconis and the late Ron Graham.