Enumerative Theory for the Tsetlin Library
Sourav Chatterjee, Persi Diaconis, Gene B. Kim

TL;DR
This paper provides an enumerative analysis of the Tsetlin library Markov chain, resolving open questions about distributions of top and bottom cards and extending results to hyperplane arrangement chambers.
Contribution
It introduces a new enumerative framework for the Tsetlin library and related hyperplane arrangements, answering longstanding open questions.
Findings
Distribution of top k cards determined
Distribution of bottom k cards characterized
Extension to hyperplane arrangement chambers achieved
Abstract
The Tsetlin library is a well-studied Markov chain on the symmetric group . It has stationary distribution the Luce model, a nonuniform distribution on , which appears in psychology, horse race betting, and tournament poker. Simple enumerative questions, such as ``what is the distribution of the top cards?'' or ``what is the distribution of the bottom cards?'' are long open. We settle these questions and draw attention to a host of parallel questions on the extension to the chambers of a hyperplane arrangement.
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Mathematical Dynamics and Fractals · Markov Chains and Monte Carlo Methods
