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 Poznanovi\'{c}.