Card shuffling and P-partitions

Abstract

In this expository article, we highlight the direct connection between card shuffling and the functions known as P-partitions that come from algebraic combinatorics. While many (but not all) of the results we discuss are known, we give a unified treatment. The key idea is this: the probability of obtaining a permutation π from shelf shuffling is the probability that a random P-partition is sorted by π, and the probability of obtaining π from riffle shuffling is the probability that a random P-partition is sorted by π−1.