$q$-enumeration of type B and D Eulerian polynomials based on parity of descents

TL;DR

This paper develops new $q$-analogues of Eulerian polynomials for types B and D, incorporating parity and inversion statistics, and derives related generating functions and recurrence relations.

Contribution

It extends previous work by introducing $q$-analogues for type B and D Eulerian polynomials with refined permutation statistics and provides new generating functions and recurrences.

Findings

Established $q$-analogues for type B and D Eulerian polynomials.

Derived generating functions for type B and D alternating descent polynomials.

Obtained $q$-analogues of Hyatt's recurrences and symmetry relations.

Abstract

Carlitz and Scoville in 1973 considered a four variable polynomial that enumerates permutations in $\mathfrak{S}_n$ with respect to the parity of its descents and ascents. In recent work, Pan and Zeng proved a $q$-analogue of Carlitz-Scoville's generating function by enumerating permutations with the above four statistice along with the inversion number. Further, they also proved a type B analogue by enumerating signed permutations with respect to the parity of descents and ascents. In this work we prove a $q$-analogue of the type B result of Pan and Zeng by enumerating permutations in $\mathfrak{B}_n$ with the above four statistics and the type B inversion number. We also obtain a $q$-analogue of the generating function for the type B bivariate alternating descent polynomials. We consider a similar five-variable polynomial in the type D Coxeter groups as well and give their egf. Alternating descents for the type D groups were previously also defined by Remmel, but our definition is slightly different. As a by-product of our proofs, we get bivariate $q$-analogues of Hyatt's recurrences for the type B and type D Eulerian polynomials. Further corollaries of our results are some symmetry relations for these polynomials and $q$-analogues of generating functions for snakes of types B and D.