The sorting index on colored permutations and even-signed permutations

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 Poznanović.

28 pages, 2 figures. Some typos are corrected in this version