On the descent polynomial of signed multipermutations
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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.References
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Additional Information
- Zhicong Lin
- Affiliation: Institut Camille Jordan, Université Claude Bernard Lyon 1, France
- Address at time of publication: School of Sciences, Jimei University, Xiamen, 361021, People’s Republic of China
- Email: lin@math.univ-lyon1.fr
- Received by editor(s): November 2, 2013
- Received by editor(s) in revised form: February 13, 2014, March 12, 2014, and June 4, 2014
- Published electronically: March 18, 2015
- Communicated by: Jim Haglund
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 3671-3685
- MSC (2010): Primary 05A05, 05A15, 01A19
- DOI: https://doi.org/10.1090/S0002-9939-2015-12555-5
- MathSciNet review: 3359561