Enumerative Theory for the Tsetlin Library

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.