A Carlitz identity for the wreath product Cᵣ ≀ Sn

Abstract

We present in this work a new flag major index fmajᵣ for the wreath product Gᵣ,n = Cᵣ ≀ Sn , where Cᵣ is the cyclic group of order r and Sn is the symmetric group on n letters. We prove that fmajᵣ is equidistributed with the length function on Gᵣ,n and that the generating function of the pair (desᵣ,fmajᵣ) over Gᵣ,n, where desᵣ is the usual descent number on Gᵣ,n, satisfies a “natural” Carlitz identity, thus unifying and generalizing earlier results due to Carlitz (in the type A case), and Chow and Gessel (in the type B case). A q-Worpitzky identity, a convolution-type recurrence and a q-Frobenius formula are also presented, with combinatorial interpretation given to the expansion coefficients of the latter formula.