A descent-excedance correspondence in colored permutation groups

It is well known that descents and excedances are equidistributed in the symmetric group.We show that the descent and excedance enumerators, summed over permutations with a fixed first letter, are identical when we perform a simple change of the first letter.We generalize this to type B and colored permutation groups.We find q-analogues of generating functions for descent enumerating polynomials with respect to the major index when summed over permutations that start with a fixed first letter.Using the flag major index, we also generalise to obtain type B results.In the univariate case, we show that the aforementioned polynomials are real-rooted but need not be palindromic.Finally, we show that a symmetrised descent enumerating polynomial over the hyperoctahedral group with fixed first letter is palindromic and gamma-positive.