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

Hiranya Kishore DeyUmesh ShankarSivaramakrishnan Sivasubramanian
Nov 2022
摘要
Carlitz and Scoville in 1973 considered four-variable polynomials enumeratingthe permutations according to the parity of both descents and ascents. In arecent work, Pan and Zeng proved a $q$-analogue of Carlitz-Scoville'sgenerating function by counting the inversion number. Moreover, they alsoproved a type B analogue by enumerating the signed permutations with respect tothe parity of descent and ascent position. In this work we prove a $q$-analogueof 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 bivariatealternating descent polynomials. Similar results are also obtained for type DCoxeter groups. As a by-product of our proofs, we get $q$-analogues of Hyatt'srecurrences for the type B Eulerian polynomials.
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