The sorting index on colored permutations and even-signed permutations

We define a new statistic sorsor on the set of colored permutations Gr,nGr,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: (ℓ,Rmil0,Rmil1,…,Rmilr−1ℓ,Rmil0,Rmil1,…,Rmilr−1, Lmil0,Lmil1,…,Lmilr−1Lmil0,Lmil1,…,Lmilr−1, Lmal0,Lmal1,…,Lmalr−1Lmal0,Lmal1,…,Lmalr−1, Lmap0,Lmap1,…,Lmapr−1Lmap0,Lmap1,…,Lmapr−1) and (sor,Cyc0,Cycr−1,…,Cyc1sor,Cyc0,Cycr−1,…,Cyc1, Lmic0,Lmicr−1,…,Lmic1Lmic0,Lmicr−1,…,Lmic1, Lmal0,Lmal1,…,Lmalr−1Lmal0,Lmal1,…,Lmalr−1, Lmap0,Lmap1,…,Lmapr−1Lmap0,Lmap1,…,Lmapr−1). Analogous results are also obtained for Coxeter group of type D. Our work generalizes recent results of Petersen, Chen–Gong–Guo and Poznanović.