Xiaoqin Gao, Frank Z. K. Li, Lingli Wan, Jane Y. X. Yang: Lattice paths related to descents and inverse descents in hyperoctahedral groups, 119-147

Abstract:

The elements in the hyperoctahedral group ${\mathfrakB}_n$ 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 $\pi\in{\mathfrakB}_n$, let $\operatorname{des}^B(\pi)$ and ${\operatorname{ides}}^B(\pi)$ be the number of descents and inverse descents in $\pi$ under the natural order, respectively, 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, respectively. Visontai algebraically studied the joint distribution of $(\operatorname{des}^B,{\operatorname{ides}}^B)$. In this paper, we introduce a new class of lattice paths on permutation grids that give a combinatorial proof for the recurrence formula of $(\operatorname{des}^B,{\operatorname{ides}}^B)$ arising from the work of Visontai. And by modifying certain combinatorial structures from the natural order to the $r$-order, we acquire a recurrence formula of the joint distribution of $(\operatorname{des}_B,{\operatorname{ides}}_B)$. As a conclusion, we find that $(\operatorname{des}^B,{\operatorname{ides}}^B)$ and $(\operatorname{des}_B,{\operatorname{ides}}_B)$ are equidistributed over the hyperoctahedral group.

Key Words: Hyperoctahedral group, inverse descents, signed permutation grids, recurrence formulas.

2020 Mathematics Subject Classification: 05A05, 05A19, 05E16.

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