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

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 $-1{<}_r-2{<}_r\cdots {<}_{r}0{<}_{r}1{<}_{r}2{<}_r\cdots$. For any $\pi \in {\mathfrak{B}}_n$, let ${\mathrm{des}}^B(\pi )$ and ${\mathrm{ides}}^B(\pi )$ be the number of descents and inverse descents in $\pi$ under the natural order, and let ${\mathrm{des}}_B(\pi )$ and ${\mathrm{ides}}_B(\pi )$ 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 $({\mathrm{des}}^B,{\mathrm{ides}}^B)$ and $({\mathrm{des}}_B,{\mathrm{ides}}_B)$ are equidistributed over both ${\mathfrak{B}}_n$ and ${\mathcal{I}}_n^B$, but not on ${\mathcal{J}}_n^B$.

41 pages