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UNIMODALITY AND COLOURED HOOK FACTORISATION

Part of: Combinatorics

Published online by Cambridge University Press:  11 November 2015

ZHICONG LIN
ZHICONG LIN*
Affiliation:
School of Science, Jimei University, Xiamen 361021, PR China CAMP, National Institute for Mathematical Sciences, Daejeon 305-811, Republic of Korea email [email protected]
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Abstract

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We prove the unimodality of some coloured $q$-Eulerian polynomials, which involve the flag excedances, the major index and the fixed points on coloured permutation groups, via two recurrence formulas. In particular, we confirm a recent conjecture of Mongelli about the unimodality of the flag excedances over type B derangements. Furthermore, we find the coloured version of Gessel’s hook factorisation, which enables us to interpret these two recurrences combinatorially. We also provide a combinatorial proof of a symmetric and unimodal expansion for the coloured derangement polynomial, which was first established by Shin and Zeng using continued fractions.

Keywords

coloured permutationsderangementsEulerian polynomialsunimodality𝛾-positivityhook factorisation

MSC classification

Primary: 05A05: Permutations, words, matrices
Secondary: 05A15: Exact enumeration problems, generating functions 05A19: Combinatorial identities, bijective combinatorics
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 93 , Issue 1 , February 2016 , pp. 1 - 12
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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