On the combinatorics of descents and inverse descents in the hyperoctahedral group

TL;DR

This paper investigates the distribution of descents and inverse descents in signed permutations within the hyperoctahedral group, providing new combinatorial proofs for recurrence formulas and revealing distributional symmetries.

Contribution

It offers combinatorial proofs for six recurrence formulas related to descents and inverse descents in hyperoctahedral groups, including new formulas and insights into their distributional properties.

Findings

Six recurrence formulas for joint distributions are proved combinatorially.

Some formulas are new, others connect to algebraic results by Visontai, Moustakas, and Cao-Liu.

Distributions of descent pairs are equal on certain subsets but differ on fixed-point free involutions.

Abstract

The elements in the hyperoctahedral group $\mathfrak{B}_n$ can be treated as signed permutations with the natural order $\cdots<-2<-1<0<1<2<\cdots$, or as colored permutations with the $r$-order $-1<_r-2<_r\cdots<_r0<_r1<_r2<_r\cdots$. For any $\pi\in\mathfrak{B}_n$, let $\operatorname{des}^B(\pi)$ and $\operatorname{ides}^B(\pi)$ be the number of descents and inverse descents in $\pi$ under the natural order, and let $\operatorname{des}_B(\pi)$ and $\operatorname{ides}_B(\pi)$ be the number of descents and inverse descents in $\pi$ under the $r$-order. In this paper, by investigating signed permutation grids under both the natural order and the $r$-order, we give combinatorial proofs for six recurrence formulas of the joint distribution of descents and inverse descents over the hyperoctahedral group $\mathfrak{B}_n$, the set in involutions of $\mathfrak{B}_n$ denoted by $\mathcal{I}_n^B$, and the set of fixed-point free involutions in $\mathfrak{B}_n$ denoted by $\mathcal{J}_n^B$, respectively. Some of these six formulas are new, and some reveal the combinatorial essences of the results obtained by Visontai, Moustakas and Cao-Liu through algebraic approaches such as quasisymmetric functions. Furthermore, from these formulas, we conclude that $(\operatorname{des}^B,\operatorname{ides}^B)$ and $(\operatorname{des}_B,\operatorname{ides}_B)$ are equidistributed over both $\mathfrak{B}_n$ and $\mathcal{I}_n^B$, but not on $\mathcal{J}_n^B$.