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

Hiranya Kishore Dey, Umesh Shankar, Sivaramakrishnan Sivasubramanian

Published 2022-11-28Version 1

Carlitz and Scoville in 1973 considered four-variable polynomials enumerating the permutations according to the parity of both descents and ascents. In a recent work, Pan and Zeng proved a $q$-analogue of Carlitz-Scoville's generating function by counting the inversion number. Moreover, they also proved a type B analogue by enumerating the signed permutations with respect to the parity of descent and ascent position. In this work we prove a $q$-analogue of the type B result of Pan and Zeng by counting the type $B$ inversion number. We also obtain a $q$-analogue of the generating functions for the bivariate alternating descent polynomials. Similar results are also obtained for type D Coxeter groups. As a by-product of our proofs, we get $q$-analogues of Hyatt's recurrences for the type B Eulerian polynomials.