Statistics on wreath products, perfect matchings, and signed words

We introduce a natural extension of Adin, Brenti, and Roichman’s major-index statistic nmaj on signed permutations (Adv. Appl. Math. 27, (2001), 210−244) to wreath products of a cyclic group with the symmetric group. We derive “insertion lemmas” which allow us to give simple bijective proofs that our extension has the same distribution as another statistic on wreath products introduced by Adin and Roichman (Europ. J. Combin. 22, (2001), 431− 446) called the flag major index. We also use our insertion lemmas to show that nmaj, the flag major index, and an inversion statistic have the same distribution on a subset of signed permutations in bijection with perfect matchings. We show that this inversion statistic has an interpretation in terms of q-counting rook placements on a shifted Ferrers board. Many results on permutation statistics extend to results on multiset permutations (words). We derive a number of analogous results for signed words, and also words with higher order roots of unity attached to them.