Enumerative Theory for the Tsetlin Library

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 $S_n$. It has stationary distribution $\pi(\sigma)$ the Luce model, a nonuniform distribution on $S_n$, which appears in psychology, horse race betting, and tournament poker. Simple enumerative questions, such as ``what is the distribution of the top $k$ cards?'' or ``what is the distribution of the bottom $k$ 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.