On the joint distribution of descents and signs of permutations

TL;DR

This paper investigates the joint distribution of descents and signs in permutations of types A and B, providing generating functions and proving central limit theorems, with applications to card shuffling randomness.

Contribution

It introduces new generating functions for the joint distribution of descents and signs, and offers two proofs of central limit theorems for these distributions in Coxeter groups.

Findings

Sign is nearly random after one riffle shuffle for large decks.

Derived explicit generating functions for Eulerian distributions with sign considerations.

Proved central limit theorems for positive and negative Eulerian numbers.

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.