A Sign-Reversing Involution on Bimahonian Generating Functions

Abstract

We explicitly evaluate the generating functions for joint distributions of pairs of the permutation statistics \(\text{inv}, {maj}\), and \({ch}\) over the symmetric group when both variables are set to \(-1\). We give a combinatorial proof by means of a sign-reversing involution that specializing the variables to \(-1\) in these bimahonian generating functions gives the number of two-colored permutations up to sign.