The sorting index on colored permutations and even-signed permutations

TL;DR

This paper introduces a new statistic on colored permutations, demonstrating its distribution matches the length function, and establishes joint equidistribution of various set-valued statistics for restricted permutations related to non-attacking rooks on Ferrers shapes, extending prior work.

Contribution

It defines a new sorting index on colored permutations and proves its distribution matches the length function, also establishing joint equidistribution results for restricted permutations and Coxeter groups of type D.

Findings

The new statistic $ ext{sor}$ has the same distribution as the length function.

Joint equidistribution of set-valued statistics for restricted colored permutations.

Results extend previous findings by Petersen, Chen-Gong-Guo, and Poznanović.

Abstract

We define a new statistic $\mathsf{sor}$ on the set of colored permutations $\mathsf{G}_{r,n}$ and prove that it has the same distribution as the length function. For the set of restricted colored permutations corresponding to the arrangements of $n$ non-attacking rooks on a fixed Ferrers shape we show that the following two sequences of set-valued statistics are joint equidistributed: $(\ell,\mathsf{Rmil}^0,\mathsf{Rmil}^1,...,\mathsf{Rmil}^{r-1}$, $\mathsf{Lmil}^0,\mathsf{Lmil}^1,...,\mathsf{Lmil}^{r-1}$, $\mathsf{Lmal}^0,\mathsf{Lmal}^1,...,\mathsf{Lmal}^{r-1}$, $\mathsf{Lmap}^0,\mathsf{Lmap}^1,...,\mathsf{Lmap}^{r-1})$ and $(\mathsf{sor},\mathsf{Cyc}^0,\mathsf{Cyc}^{r-1},...,\mathsf{Cyc}^{1}$, $\mathsf{Lmic}^0,\mathsf{Lmic}^{r-1},...,\mathsf{Lmic}^{1}$, $\mathsf{Lmal}^0,\mathsf{Lmal}^1,...,\mathsf{Lmal}^{r-1}$, $\mathsf{Lmap}^0,\mathsf{Lmap}^1,...,\mathsf{Lmap}^{r-1})$. Analogous results are also obtained for Coxeter group of type $D$. Our results extend recent results of Petersen, Chen-Gong-Guo and Poznanovi\'{c}.