Abstract
We study the joint distribution of descents and sign for elements of the symmetric group and the hyperoctahedral group (Coxeter groups of types $A$ and $B$). For both groups, this has an application to riffle shuffling: for large decks of cards the sign is close to random after a single shuffle. In both groups, we derive generating functions for the Eulerian distribution refined according to sign, and use them to give two proofs of central limit theorems for positive and negative Eulerian numbers.
Highlights
The distribution of descents over all permutations is known as the Eulerian distribution, and the number of permutations of n elements with a given number of descents is known as an Eulerian number
The Eulerian numbers are ubiquitous in combinatorics; see [23]
We provide some definitions in the symmetric group case
Summary
The distribution of descents over all permutations is known as the Eulerian distribution, and the number of permutations of n elements with a given number of descents is known as an Eulerian number. Desarménien and Foata [9], who studied “signed” Eulerian numbers The symmetric and hyperoctahedral groups are Coxeter groups of types An−1 and Bn , and the distribution of descents is well-understood in any finite Coxeter group; see, e.g., the recent paper of Kahle and Stump [20] for the current state of knowledge. To our knowledge the limiting distribution of descents and sign has not been investigated outside of the symmetric group, and this paper is the first to make connections between such distributions and card shuffling.
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