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On the descent polynomial of signed multipermutations


Author: Zhicong Lin
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
Published electronically: March 18, 2015
MathSciNet review: 3359561
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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.


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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

DOI: https://doi.org/10.1090/S0002-9939-2015-12555-5
Keywords: Descents, ascents, inversion sequences, signed multipermutations, real-rootedness
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
Article copyright: © Copyright 2015 American Mathematical Society