Pub Date : 2024-02-06DOI: 10.1007/s00026-023-00683-x
Ezgi Kantarcı Oǧuz, Cem Yalım Özel, Mohan Ravichandran
We introduce a class of polytopes that we call chainlink polytopes and show that they allow us to construct infinite families of pairs of non-isomorphic rational polytopes with the same Ehrhart quasipolynomial. Our construction is based on circular fence posets, a recently introduced class of posets, which admit a non-obvious and nontrivial symmetry in their rank sequences. We show that this symmetry can be lifted to the level of polyhedral models (which we call chainlink polytopes) for these posets. Along the way, we introduce the related class of chainlink posets and show that they exhibit analogous nontrivial symmetry properties. We further prove an outstanding conjecture on the unimodality of rank polynomials of circular fence posets.
{"title":"Chainlink Polytopes and Ehrhart Equivalence","authors":"Ezgi Kantarcı Oǧuz, Cem Yalım Özel, Mohan Ravichandran","doi":"10.1007/s00026-023-00683-x","DOIUrl":"10.1007/s00026-023-00683-x","url":null,"abstract":"<div><p>We introduce a class of polytopes that we call chainlink polytopes and show that they allow us to construct infinite families of pairs of non-isomorphic rational polytopes with the same Ehrhart quasipolynomial. Our construction is based on circular fence posets, a recently introduced class of posets, which admit a non-obvious and nontrivial symmetry in their rank sequences. We show that this symmetry can be lifted to the level of polyhedral models (which we call chainlink polytopes) for these posets. Along the way, we introduce the related class of chainlink posets and show that they exhibit analogous nontrivial symmetry properties. We further prove an outstanding conjecture on the unimodality of rank polynomials of circular fence posets.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"28 4","pages":"1141 - 1166"},"PeriodicalIF":0.6,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139772175","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this work, we give the sharp upper bound for the number of cliques in graphs with bounded odd circumferences. This generalized Turán-type result is an extension of the celebrated Erdős and Gallai theorem and a strengthening of Luo’s recent result. The same bound for graphs with bounded even circumferences is a trivial application of the theorem of Li and Ning.
{"title":"The Maximum Number of Cliques in Graphs with Bounded Odd Circumference","authors":"Zequn Lv, Ervin Győri, Zhen He, Nika Salia, Chuanqi Xiao, Xiutao Zhu","doi":"10.1007/s00026-023-00682-y","DOIUrl":"10.1007/s00026-023-00682-y","url":null,"abstract":"<div><p>In this work, we give the sharp upper bound for the number of cliques in graphs with bounded odd circumferences. This generalized Turán-type result is an extension of the celebrated Erdős and Gallai theorem and a strengthening of Luo’s recent result. The same bound for graphs with bounded even circumferences is a trivial application of the theorem of Li and Ning.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"28 4","pages":"1119 - 1125"},"PeriodicalIF":0.6,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00026-023-00682-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139578299","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-23DOI: 10.1007/s00026-023-00684-w
Andrew Y. Z. Wang, Ang Xiao
There has been a tremendous amount of research on the truncated theta series in the past decade. How can we understand them combinatorially? In this paper, we investigate the truncated theorems of three classical theta series of Euler and Gauss, and provide a unified combinatorial treatment. Meanwhile, we propose a possible and more direct approach to deal with these truncated theorems.
{"title":"A Unified Combinatorial Treatment for Three Classical Truncated Theta Series","authors":"Andrew Y. Z. Wang, Ang Xiao","doi":"10.1007/s00026-023-00684-w","DOIUrl":"https://doi.org/10.1007/s00026-023-00684-w","url":null,"abstract":"<p>There has been a tremendous amount of research on the truncated theta series in the past decade. How can we understand them combinatorially? In this paper, we investigate the truncated theorems of three classical theta series of Euler and Gauss, and provide a unified combinatorial treatment. Meanwhile, we propose a possible and more direct approach to deal with these truncated theorems.</p>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"7 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139558141","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-02DOI: 10.1007/s00026-023-00681-z
Qiuyu Ren, Shengtong Zhang
We provide a much shorter proof of Defant and Kravitz’s theorem that the length of Hitomezashi loops is congruent to 4 modulo 8. Our novel idea is to consider the length module 8 for Hitomezashi paths that take an excursion in a half-plane region.
{"title":"A Succinct Proof of Defant and Kravitz’s Theorem on the Length of Hitomezashi Loops","authors":"Qiuyu Ren, Shengtong Zhang","doi":"10.1007/s00026-023-00681-z","DOIUrl":"https://doi.org/10.1007/s00026-023-00681-z","url":null,"abstract":"<p>We provide a much shorter proof of Defant and Kravitz’s theorem that the length of Hitomezashi loops is congruent to 4 modulo 8. Our novel idea is to consider the length module 8 for Hitomezashi paths that take an excursion in a half-plane region.</p>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"21 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139096718","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-22DOI: 10.1007/s00026-023-00679-7
Soichi Okada, Akiyoshi Tsuchiya
Stanley introduced and studied two lattice polytopes, the order polytope and chain polytope, associated with a finite poset. Recently, Ohsugi and Tsuchiya introduce an enriched version of them, called the enriched order polytope and enriched chain polytope. In this paper, we give a piecewise-linear bijection between these enriched poset polytopes, which is an enriched analogue of Stanley’s transfer map and bijectively proves that they have the same Ehrhart polynomials. Also, we construct explicitly unimodular triangulations of two enriched poset polytopes, which are the order complexes of graded posets.
{"title":"Two Enriched Poset Polytopes","authors":"Soichi Okada, Akiyoshi Tsuchiya","doi":"10.1007/s00026-023-00679-7","DOIUrl":"10.1007/s00026-023-00679-7","url":null,"abstract":"<div><p>Stanley introduced and studied two lattice polytopes, the order polytope and chain polytope, associated with a finite poset. Recently, Ohsugi and Tsuchiya introduce an enriched version of them, called the enriched order polytope and enriched chain polytope. In this paper, we give a piecewise-linear bijection between these enriched poset polytopes, which is an enriched analogue of Stanley’s transfer map and bijectively proves that they have the same Ehrhart polynomials. Also, we construct explicitly unimodular triangulations of two enriched poset polytopes, which are the order complexes of graded posets.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"28 1","pages":"257 - 282"},"PeriodicalIF":0.6,"publicationDate":"2023-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139031804","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-21DOI: 10.1007/s00026-023-00678-8
Péter L. Erdős, Tamás Róbert Mezei, István Miklós
The approximate uniform sampling of graph realizations with a given degree sequence is an everyday task in several social science, computer science, engineering etc. projects. One approach is using Markov chains. The best available current result about the well-studied switch Markov chain is that it is rapidly mixing on P-stable degree sequences (see DOI:10.1016/j.ejc.2021.103421). The switch Markov chain does not change any degree sequence. However, there are cases where degree intervals are specified rather than a single degree sequence. (A natural scenario where this problem arises is in hypothesis testing on social networks that are only partially observed.) Rechner, Strowick, and Müller–Hannemann introduced in 2018 the notion of degree interval Markov chain which uses three (separately well studied) local operations (switch, hinge-flip and toggle), and employing on degree sequence realizations where any two sequences under scrutiny have very small coordinate-wise distance. Recently, Amanatidis and Kleer published a beautiful paper (DOI:10.4230/LIPIcs.STACS.2023.7), showing that the degree interval Markov chain is rapidly mixing if the sequences are coming from a system of very thin intervals which are centered not far from a regular degree sequence. In this paper, we substantially extend their result, showing that the degree interval Markov chain is rapidly mixing if the intervals are centered at P-stable degree sequences.
摘要 在一些社会科学、计算机科学、工程学等项目中,对具有给定度序列的图形现实进行近似均匀采样是一项日常任务。一种方法是使用马尔可夫链。关于研究得很透彻的开关马尔可夫链,目前最好的结果是它能在 P 个稳定的度序列上快速混合(见 DOI:10.1016/j.ejc.2021.103421)。切换马尔可夫链不会改变任何度序列。然而,在有些情况下,指定的是度数区间而不是单一的度数序列。(出现这个问题的一个自然场景是对只有部分观测数据的社交网络进行假设检验)。Rechner, Strowick 和 Müller-Hannemann 在 2018 年提出了程度区间马尔科夫链的概念,它使用了三种(分别研究得很好)局部操作(切换、铰链-翻转和切换),并采用了程度序列实现,其中任何两个被审查的序列都具有非常小的坐标距离。最近,Amanatidis 和 Kleer 发表了一篇漂亮的论文(DOI:10.4230/LIPIcs.STACS.2023.7),表明如果序列来自一个非常细的区间系统,而这些区间的中心离一个规则的度数序列不远,那么度数区间马尔可夫链就会迅速混合。在本文中,我们大幅扩展了他们的结果,证明了如果区间以 P 个稳定的度序列为中心,度区间马尔可夫链是快速混合的。
{"title":"Approximate Sampling of Graphs with Near-P-Stable Degree Intervals","authors":"Péter L. Erdős, Tamás Róbert Mezei, István Miklós","doi":"10.1007/s00026-023-00678-8","DOIUrl":"10.1007/s00026-023-00678-8","url":null,"abstract":"<div><p>The approximate uniform sampling of graph realizations with a given degree sequence is an everyday task in several social science, computer science, engineering etc. projects. One approach is using Markov chains. The best available current result about the well-studied switch Markov chain is that it is rapidly mixing on <i>P</i>-stable degree sequences (see <span>DOI:10.1016/j.ejc.2021.103421</span>). The switch Markov chain does not change any degree sequence. However, there are cases where degree intervals are specified rather than a single degree sequence. (A natural scenario where this problem arises is in hypothesis testing on social networks that are only partially observed.) Rechner, Strowick, and Müller–Hannemann introduced in 2018 the notion of <i>degree interval Markov chain</i> which uses three (separately well studied) local operations (switch, hinge-flip and toggle), and employing on degree sequence realizations where any two sequences under scrutiny have very small coordinate-wise distance. Recently, Amanatidis and Kleer published a beautiful paper (<span>DOI:10.4230/LIPIcs.STACS.2023.7</span>), showing that the degree interval Markov chain is rapidly mixing if the sequences are coming from a system of very thin intervals which are centered not far from a regular degree sequence. In this paper, we substantially extend their result, showing that the degree interval Markov chain is rapidly mixing if the intervals are centered at <i>P</i>-stable degree sequences.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"28 1","pages":"223 - 256"},"PeriodicalIF":0.6,"publicationDate":"2023-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00026-023-00678-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139017850","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-21DOI: 10.1007/s00026-023-00675-x
Vuong Bui
Let P(n) be the number of polyominoes of n cells and (lambda ) be Klarner’s constant, that is, (lambda =lim _{nrightarrow infty } root n of {P(n)}). We show that there exist some positive numbers A, T, so that for every n
$$begin{aligned} P(n) ge An^{-Tlog n} lambda ^n. end{aligned}$$
This is somewhat a step toward the well-known conjecture that there exist positive (C,theta ), so that (P(n)sim Cn^{-theta }lambda ^n) for every n. In fact, if we assume another popular conjecture that (P(n)/P(n-1)) is increasing, we can get rid of (log n) to have
Beside the above theoretical result, we also conjecture that the ratio of the number of some class of polyominoes, namely inconstructible polyominoes, over P(n) is decreasing, by observing this behavior for the available values. The conjecture opens a nice approach to bounding (lambda ) from above, since if it is the case, we can conclude that
$$begin{aligned} lambda < 4.1141, end{aligned}$$
which is quite close to the current best lower bound (lambda > 4.0025) and greatly improves the current best upper bound (lambda < 4.5252). The approach is merely analytically manipulating the known or likely properties of the function P(n), instead of giving new insights of the structure of polyominoes. The techniques can be applied to other lattice animals and self-avoiding polygons of a given area with almost no change.
{"title":"An Asymptotic Lower Bound on the Number of Polyominoes","authors":"Vuong Bui","doi":"10.1007/s00026-023-00675-x","DOIUrl":"10.1007/s00026-023-00675-x","url":null,"abstract":"<div><p>Let <i>P</i>(<i>n</i>) be the number of polyominoes of <i>n</i> cells and <span>(lambda )</span> be Klarner’s constant, that is, <span>(lambda =lim _{nrightarrow infty } root n of {P(n)})</span>. We show that there exist some positive numbers <i>A</i>, <i>T</i>, so that for every <i>n</i></p><div><div><span>$$begin{aligned} P(n) ge An^{-Tlog n} lambda ^n. end{aligned}$$</span></div></div><p>This is somewhat a step toward the well-known conjecture that there exist positive <span>(C,theta )</span>, so that <span>(P(n)sim Cn^{-theta }lambda ^n)</span> for every <i>n</i>. In fact, if we assume another popular conjecture that <span>(P(n)/P(n-1))</span> is increasing, we can get rid of <span>(log n)</span> to have </p><div><div><span>$$begin{aligned} P(n)ge An^{-T}lambda ^n. end{aligned}$$</span></div></div><p>Beside the above theoretical result, we also conjecture that the ratio of the number of some class of polyominoes, namely inconstructible polyominoes, over <i>P</i>(<i>n</i>) is decreasing, by observing this behavior for the available values. The conjecture opens a nice approach to bounding <span>(lambda )</span> from above, since if it is the case, we can conclude that </p><div><div><span>$$begin{aligned} lambda < 4.1141, end{aligned}$$</span></div></div><p>which is quite close to the current best lower bound <span>(lambda > 4.0025)</span> and greatly improves the current best upper bound <span>(lambda < 4.5252)</span>. The approach is merely analytically manipulating the known or likely properties of the function <i>P</i>(<i>n</i>), instead of giving new insights of the structure of polyominoes. The techniques can be applied to other lattice animals and self-avoiding polygons of a given area with almost no change.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"28 2","pages":"459 - 484"},"PeriodicalIF":0.6,"publicationDate":"2023-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139018413","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-15DOI: 10.1007/s00026-023-00680-0
Gregg Musiker, Son Nguyen
We study labeled chip-firing on binary trees starting with (2^n-1) chips initially placed at the root. We prove a sorting property of terminal configurations of the process. We also analyze the end game moves poset and prove that this poset is a modular lattice.
{"title":"Labeled Chip-Firing on Binary Trees with (2^n-1) Chips","authors":"Gregg Musiker, Son Nguyen","doi":"10.1007/s00026-023-00680-0","DOIUrl":"10.1007/s00026-023-00680-0","url":null,"abstract":"<div><p>We study labeled chip-firing on binary trees starting with <span>(2^n-1)</span> chips initially placed at the root. We prove a sorting property of terminal configurations of the process. We also analyze the end game moves poset and prove that this poset is a modular lattice.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"28 4","pages":"1167 - 1197"},"PeriodicalIF":0.6,"publicationDate":"2023-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138689293","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-13DOI: 10.1007/s00026-023-00674-y
Petr Gregor, Arturo Merino, Torsten Mütze
We say that a Hamilton cycle (C=(x_1,ldots ,x_n)) in a graph G is k-symmetric, if the mapping (x_imapsto x_{i+n/k}) for all (i=1,ldots ,n), where indices are considered modulo n, is an automorphism of G. In other words, if we lay out the vertices (x_1,ldots ,x_n) equidistantly on a circle and draw the edges of G as straight lines, then the drawing of G has k-fold rotational symmetry, i.e., all information about the graph is compressed into a (360^circ /k) wedge of the drawing. The maximum k for which there exists a k-symmetric Hamilton cycle in G is referred to as the Hamilton compression ofG. We investigate the Hamilton compression of four different families of vertex-transitive graphs, namely hypercubes, Johnson graphs, permutahedra and Cayley graphs of abelian groups. In several cases, we determine their Hamilton compression exactly, and in other cases, we provide close lower and upper bounds. The constructed cycles have a much higher compression than several classical Gray codes known from the literature. Our constructions also yield Gray codes for bitstrings, combinations and permutations that have few tracks and/or that are balanced.
如果对于所有 (i=1,ldots,n)(这里的索引都是以 n 为模的),映射 (x_imapstox_{i+n/k})是 G 的自动变形,那么我们就说图 G 中的汉密尔顿循环 (C=(x_1,ldots ,x_n))是 k 对称的。换句话说,如果我们把顶点 (x_1,ldots ,x_n) 等距地画在一个圆上,并把 G 的边画成直线,那么 G 的画法就具有 k 倍旋转对称性,也就是说、关于图形的所有信息都被压缩到了绘图的一个 (360^circ /k)楔形中。我们研究了四个不同顶点变换图族的汉密尔顿压缩,它们分别是超立方体图、约翰逊图、永恒面图和无性群的卡莱图。在几种情况下,我们精确地确定了它们的汉密尔顿压缩率,在其他情况下,我们提供了接近的下限和上限。所构建的循环比文献中已知的几种经典格雷码的压缩率要高得多。我们的构造还产生了位串、组合和排列的灰色代码,这些代码的轨迹很少,而且/或者是平衡的。
{"title":"The Hamilton Compression of Highly Symmetric Graphs","authors":"Petr Gregor, Arturo Merino, Torsten Mütze","doi":"10.1007/s00026-023-00674-y","DOIUrl":"10.1007/s00026-023-00674-y","url":null,"abstract":"<div><p>We say that a Hamilton cycle <span>(C=(x_1,ldots ,x_n))</span> in a graph <i>G</i> is <i>k</i>-symmetric, if the mapping <span>(x_imapsto x_{i+n/k})</span> for all <span>(i=1,ldots ,n)</span>, where indices are considered modulo <i>n</i>, is an automorphism of <i>G</i>. In other words, if we lay out the vertices <span>(x_1,ldots ,x_n)</span> equidistantly on a circle and draw the edges of <i>G</i> as straight lines, then the drawing of <i>G</i> has <i>k</i>-fold rotational symmetry, i.e., all information about the graph is compressed into a <span>(360^circ /k)</span> wedge of the drawing. The maximum <i>k</i> for which there exists a <i>k</i>-symmetric Hamilton cycle in <i>G</i> is referred to as the <i>Hamilton compression of</i> <i>G</i>. We investigate the Hamilton compression of four different families of vertex-transitive graphs, namely hypercubes, Johnson graphs, permutahedra and Cayley graphs of abelian groups. In several cases, we determine their Hamilton compression exactly, and in other cases, we provide close lower and upper bounds. The constructed cycles have a much higher compression than several classical Gray codes known from the literature. Our constructions also yield Gray codes for bitstrings, combinations and permutations that have few tracks and/or that are balanced.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"28 2","pages":"379 - 437"},"PeriodicalIF":0.6,"publicationDate":"2023-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00026-023-00674-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138689490","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-12DOI: 10.1007/s00026-023-00677-9
Tewodros Amdeberhan, Stephan Wagner
Let (nge 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.
{"title":"The MacMahon q-Catalan is Convex","authors":"Tewodros Amdeberhan, Stephan Wagner","doi":"10.1007/s00026-023-00677-9","DOIUrl":"10.1007/s00026-023-00677-9","url":null,"abstract":"<div><p>Let <span>(nge 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":0.6,"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":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138579334","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}