On the descent polynomial of signed multipermutations

Lin, Zhicong

doi:10.1090/S0002-9939-2015-12555-5

On the descent polynomial of signed multipermutations
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Proc. Amer. Math. Soc. 143 (2015), 3671-3685 Request permission

Abstract:

Motivated by a conjecture of Savage and Visontai about the equidistribution of the descent statistic on signed permutations of the multiset $\{1,1,2,2,\ldots ,n,n\}$ and the ascent statistic on $(1,4,3,8,\ldots ,2n-1,4n)$-inversion sequences, we investigate the descent polynomial of the signed permutations of a general multiset (multipermutations). We obtain a factorial generating function formula for a $q$-analog of these descent polynomials and apply it to show that they have only real roots. Two different proofs of the conjecture of Savage and Visontai are provided. Furthermore, multivariate identities that enumerate two different Euler–Mahonian distributions on type B Coxeter groups due to Beck and Braun are generalized to signed multipermutations.

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