A combinatorial proof of the parity unimodality of the (m,n)-rational q-Catalan polynomial for m=3

Yue Zhou and Yueming Zhong

Applied Mathematics and Computation, 2023, vol. 440, issue C

Abstract: A sequence a0,a1,a2,⋯ is parity unimodal if it exists k1 and k2 such that a0≤a2≤⋯≤ak1≥ak1+2≥⋯ and a1≤a3≤⋯≤ak2≥ak2+2≥⋯. A polynomial f(q)=∑i=0naiqi is parity unimodal if its coefficient sequence a0,a1,⋯,an is parity unimodal. Recently, Xin and Zhong conjectured that the (m,n)-rational q-Catalan polynomial is parity unimodal. They showed that this conjecture holds for m≤5 using the constant term method and generating functions. In this paper, we give a combinatorial proof of the m=3 case.

Keywords: Rational q-Catalan polynomials; Unimodal sequences; Posets; Chain decompositions (search for similar items in EconPapers)
Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:440:y:2023:i:c:s0096300322006890

DOI: 10.1016/j.amc.2022.127616

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