Some remarks on the joint distribution of descents and inverse descents

TL;DR

This paper investigates the joint distribution of descents and inverse descents in permutations, exploring recurrence relations, generalizations, and connections to cyclic statistics, while proposing a combinatorial model.

Contribution

It develops a recurrence for the generating function coefficients and extends the analysis to type B, offering new insights and conjectures in permutation statistics.

Findings

Derived a recurrence for the coefficients of the generating function.

Extended the analysis to type B permutations.

Connected descents with cyclic and inversion sequence statistics.

Abstract

We study the joint distribution of descents and inverse descents over the set of permutations of n letters. Gessel conjectured that the two-variable generating function of this distribution can be expanded in a given basis with nonnegative integer coefficients. We investigate the action of the Eulerian operators that give the recurrence for these generating functions. As a result we devise a recurrence for the coefficients but are unable to settle the conjecture. We examine generalizations of the conjecture and obtain a type B analog of the recurrence satisfied by the two-variable generating function. We also exhibit some connections to cyclic descents and cyclic inverse descents. Finally, we propose a combinatorial model in terms of statistics on inversion sequences.