The MacMahon q-Catalan is Convex

IF 0.6 4区数学 Q4 MATHEMATICS, APPLIED

Annals of Combinatorics Pub Date : 2023-12-12 DOI:10.1007/s00026-023-00677-9

Tewodros Amdeberhan, Stephan Wagner

引用次数: 0

Abstract

Let \(n\ge 2\) be an integer. We prove the convexity of the so-called MacMahon q-Catalan polynomials \(C_n(q)=\frac{1}{[n+1]_q}\left[ 2n \atop n \right] _q\) viewed as functions of q over the entire set of reals. Along the way, several auxiliary properties of the q-Catalan polynomials and intermediate results in the form of inequalities are presented, with the aim to make the paper self-contained. We also include a commentary on the convexity of the generating function for the integer partitions.

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MacMahon q-Catalan 是凸的

让 \(n\ge 2\) 是一个整数。我们证明了所谓的 MacMahon q-Catalan 多项式 \(C_n(q)=\frac{1}{[n+1]_q}\left[ 2n \atop n \right] _q\) 作为 q 在整个实数集上的函数的凸性。在此过程中，我们以不等式的形式介绍了 q-Catalan 多项式的几个辅助性质和中间结果，目的是使本文自成体系。我们还对整数分部生成函数的凸性进行了评述。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Annals of Combinatorics 数学-应用数学

CiteScore

1.00

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Annals of Combinatorics publishes outstanding contributions to combinatorics with a particular focus on algebraic and analytic combinatorics, as well as the areas of graph and matroid theory. Special regard will be given to new developments and topics of current interest to the community represented by our editorial board. The scope of Annals of Combinatorics is covered by the following three tracks: Algebraic Combinatorics: Enumerative combinatorics, symmetric functions, Schubert calculus / Combinatorial Hopf algebras, cluster algebras, Lie algebras, root systems, Coxeter groups / Discrete geometry, tropical geometry / Discrete dynamical systems / Posets and lattices Analytic and Algorithmic Combinatorics: Asymptotic analysis of counting sequences / Bijective combinatorics / Univariate and multivariable singularity analysis / Combinatorics and differential equations / Resolution of hard combinatorial problems by making essential use of computers / Advanced methods for evaluating counting sequences or combinatorial constants / Complexity and decidability aspects of combinatorial sequences / Combinatorial aspects of the analysis of algorithms Graphs and Matroids: Structural graph theory, graph minors, graph sparsity, decompositions and colorings / Planar graphs and topological graph theory, geometric representations of graphs / Directed graphs, posets / Metric graph theory / Spectral and algebraic graph theory / Random graphs, extremal graph theory / Matroids, oriented matroids, matroid minors / Algorithmic approaches