The sorting index and set-valued joint equidistributions of $\mathcal{B}_n$ and $\mathcal{D}_n$

Abstract

The sorting indices $\text{sor}_B$ and $\text{sor}_D$ on the Coxeter groups of type $B$ and $D$ respectively are defined by Petersen and it is proved that $(\text{inv}_B, \text{rlmin})$ and $(\text{sor}_B, \ell'_B)$ have the same joint distribution for type $B$ while $\text{inv}_D$ and $\text{sor}_D$ have the same distribution for type $D$. These results, including a set-valued extension of type $B$ involving two equildistributed pairs of three statistics, are proved combinatorially by Chen et al. via two mappings $\varphi:=\text{(B-code)}^{-1}\circ \text{(A-code)}$ and $\psi:=\text{(D-code)}^{-1}\circ \text{(C-code)}$. In this paper we further extend these results. In type $B$ we prove a set-valued joint equildistribution between a pair of seven statistics, and find a five-variable generating function. In type $D$ we define new set-valued statistics, among them $\text{Cyc}^+_D$ and $\text{Cyc}^-_D$, and firstly find a set-valued joint equidistribution between a pair of five statistics and find a four-variable generating function.