Enumeration of permutations by the parity of descent positions

TL;DR

This paper develops new q-analogues and type B analogues for permutation enumeration based on the parity of descent and ascent positions, extending classical descent polynomial results.

Contribution

It introduces novel q-analogues of Carlitz-Scoville's generating function and a type B analogue, enriching the combinatorial understanding of permutations by parity.

Findings

Derived a q-analogue of Carlitz-Scoville's generating function.

Established a type B analogue for signed permutations.

Provided alternative proofs for gamma-positivity results.

Abstract

Noticing that some recent variations of descent polynomials are special cases of Carlitz and Scoville's four-variable polynomials, which enumerate permutations by the parity of descent and ascent positions, we prove a $q$-analogue of Carlitz-Scoville's generating function by counting the inversion number and a type B analogue by enumerating the signed permutations with respect to the parity of desecnt and ascent positions. As a by-product of our formulas, we obtain a $q$-analogue of Chebikin's formula for alternating descent polynomials, an alternative proof of Sun's gamma-positivity of her bivariate Eulerian polynomials and a type B analogue, the latter refines Petersen's gamma-positivity of the type B Eulerian polynomials.