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 $({\text{sor}}_B,{\ell }_B^{\prime })$ 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 $\phi :={\text{(B-code)}}^{-1}\circ \text{(A-code)}$ 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.

arXiv commentThis paper has been withdrawn by the authors because all the results are covered by authors' newly submission titled "The sorting index on colored permutations and even-signed permutations" (arXiv:1409.8093)