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 π(σ) 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.