{"title":"The MacMahon q-Catalan is Convex","authors":"Tewodros Amdeberhan, Stephan Wagner","doi":"10.1007/s00026-023-00677-9","DOIUrl":null,"url":null,"abstract":"<div><p>Let <span>\\(n\\ge 2\\)</span> be an integer. We prove the convexity of the so-called MacMahon <i>q</i>-Catalan polynomials <span>\\(C_n(q)=\\frac{1}{[n+1]_q}\\left[ 2n \\atop n \\right] _q\\)</span> viewed as functions of <i>q</i> over the entire set of reals. Along the way, several auxiliary properties of the <i>q</i>-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.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"28 3","pages":"1003 - 1019"},"PeriodicalIF":0.7000,"publicationDate":"2023-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00026-023-00677-9.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Combinatorics","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00026-023-00677-9","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
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.
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
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