A descent-excedance correspondence in colored permutation groups

TL;DR

This paper explores a new correspondence between descents and excedances in colored permutation groups, generalizing known symmetric group results and extending them to type B and other groups.

Contribution

It introduces a descent-excedance correspondence in colored permutation groups, generalizing classical results and providing new identities for type B and related groups.

Findings

Descent and excedance enumerators are identical under a simple first letter change.

The correspondence extends to type B and other colored permutation groups.

A type B version of Conger's refinement of the Carlitz identity is established.

Abstract

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 other colored permutation groups. We are led to defining descents and excedances through linear orders. With respect to a particular order, when the number of colors is even, we get a result that generalizes the type B results. Lastly, we get a type B counterpart of Conger's result which refines the well known Carlitz identity.