European Journal of Combinatorics最新文献

{"title":"First order logic and twin-width in tournaments and dense oriented graphs","authors":"Colin Geniet ,&nbsp;Stéphan Thomassé","doi":"10.1016/j.ejc.2025.104247","DOIUrl":"10.1016/j.ejc.2025.104247","url":null,"abstract":"<div><div>We characterise the classes of tournaments with tractable first-order model checking. For every hereditary class of tournaments <span><math><mi>T</mi></math></span>, first-order model checking is either fixed parameter tractable or <span><math><mrow><mtext>AW</mtext><mrow><mo>[</mo><mo>∗</mo><mo>]</mo></mrow></mrow></math></span>-hard. This dichotomy coincides with the fact that <span><math><mi>T</mi></math></span> has either bounded or unbounded twin-width, and that the growth of <span><math><mi>T</mi></math></span> is either at most exponential or at least factorial. From the model-theoretic point of view, we show that NIP classes of tournaments coincide with bounded twin-width. Twin-width is also characterised by three infinite families of obstructions: <span><math><mi>T</mi></math></span> has bounded twin-width if and only if it excludes at least one tournament from each family. This generalises results of Bonnet et al. on ordered graphs.</div><div>The key for these results is a polynomial time algorithm that takes as input a tournament <span><math><mi>T</mi></math></span> and computes a linear order <span><math><mo>&lt;</mo></math></span> on <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow></mrow></math></span> such that the twin-width of the birelation <span><math><mrow><mo>(</mo><mi>T</mi><mo>,</mo><mo>&lt;</mo><mo>)</mo></mrow></math></span> is at most some function of the twin-width of <span><math><mi>T</mi></math></span>. Since approximating twin-width can be done in polynomial time for an ordered structure <span><math><mrow><mo>(</mo><mi>T</mi><mo>,</mo><mo>&lt;</mo><mo>)</mo></mrow></math></span>, this provides a polynomial time approximation of twin-width for tournaments.</div><div>Our results extend to oriented graphs with stable sets of bounded size, which may also be augmented by arbitrary binary relations.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"132 ","pages":"Article 104247"},"PeriodicalIF":0.9,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145247904","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Flips in two-dimensional hypertriangulations","authors":"Herbert Edelsbrunner ,&nbsp;Alexey Garber ,&nbsp;Mohadese Ghafari ,&nbsp;Teresa Heiss ,&nbsp;Morteza Saghafian","doi":"10.1016/j.ejc.2025.104248","DOIUrl":"10.1016/j.ejc.2025.104248","url":null,"abstract":"<div><div>We study flips in hypertriangulations of planar points sets. Here a level-<span><math><mi>k</mi></math></span> hypertriangulation of <span><math><mi>n</mi></math></span> points in the plane is a subdivision induced by the projection of a <span><math><mi>k</mi></math></span>-hypersimplex, which is the convex hull of the barycenters of the <span><math><mrow><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></math></span>-dimensional faces of the standard <span><math><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></math></span>-simplex. In particular, we introduce four types of flips and prove that the level-2 hypertriangulations are connected by these flips.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"132 ","pages":"Article 104248"},"PeriodicalIF":0.9,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145247903","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A proof of some conjectures of Garvan on partitions rank and crank inequalities","authors":"Renrong Mao,&nbsp;Jie Huang,&nbsp;Fan Yang","doi":"10.1016/j.ejc.2025.104253","DOIUrl":"10.1016/j.ejc.2025.104253","url":null,"abstract":"<div><div>In 1988, Garvan made conjectures on inequalities satisfied by ranks and cranks modulo 5 and 7. We obtain improvements to two of these inequalities in this paper.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"132 ","pages":"Article 104253"},"PeriodicalIF":0.9,"publicationDate":"2025-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145248003","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On two conjectures of Shallit about Thue–Morse-like sequences","authors":"Lubomíra Dvořáková ,&nbsp;Savinien Kreczman ,&nbsp;Edita Pelantová","doi":"10.1016/j.ejc.2025.104250","DOIUrl":"10.1016/j.ejc.2025.104250","url":null,"abstract":"<div><div>We study a class of infinite words <span><math><msub><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>, <span><math><mrow><mi>k</mi><mo>∈</mo><mi>N</mi><mo>,</mo><mi>k</mi><mo>≥</mo><mn>1</mn></mrow></math></span>, recently introduced by J. Shallit. This class includes the Thue–Morse sequence <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>, the Fibonacci–Thue–Morse sequence <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, and the Allouche–Johnson sequence <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>. Shallit stated and for <span><math><mrow><mi>k</mi><mo>=</mo><mn>3</mn></mrow></math></span> proved two conjectures on properties of <span><math><msub><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>. The first conjecture concerns the factor complexity, the second one the critical exponent of these words. We confirm the validity of both conjectures for every <span><math><mi>k</mi></math></span>.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"132 ","pages":"Article 104250"},"PeriodicalIF":0.9,"publicationDate":"2025-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145247905","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Growth rates of permutations with given descent or peak set","authors":"Mohamed Omar,&nbsp;Justin M. Troyka","doi":"10.1016/j.ejc.2025.104246","DOIUrl":"10.1016/j.ejc.2025.104246","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Given a set &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mo&gt;⊆&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, consider the sequences &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; where for any &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; respectively count the number of permutations in the symmetric group &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; whose descent set (respectively peak set) is &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mo&gt;∩&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. We investigate the growth rates &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;gr&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;lim&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;mi&gt;∞&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mfenced&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfenced&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;gr&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;lim&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;mi&gt;∞&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mfenced&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfenced&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; over all &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mo&gt;⊆&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. Our main contributions are two-fold. Firstly, we prove that the numbers &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;gr&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; over all &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mo&gt;⊆&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; are exactly the interval &lt;span&gt;&lt;math&gt;&lt;mfenced&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mi&gt;π&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfenced&gt;&lt;/math&gt;&lt;/span&gt;. To do so, we construct an algorithm that explicitly builds &lt;span&gt;&lt;math&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; for any desired limit &lt;span&gt;&lt;math&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; in the interval. Secondly, we prove that the numbers &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;gr&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; for periodic sets &lt;span&gt;&lt;math&gt;","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"131 ","pages":"Article 104246"},"PeriodicalIF":0.9,"publicationDate":"2025-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145220359","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the biases and asymptotics of partitions with finite choices of parts","authors":"Jiyou Li,&nbsp;Sicheng Zhao","doi":"10.1016/j.ejc.2025.104245","DOIUrl":"10.1016/j.ejc.2025.104245","url":null,"abstract":"<div><div>Biases in integer partitions have been studied recently. For three disjoint subsets <span><math><mrow><mi>R</mi><mo>,</mo><mi>S</mi><mo>,</mo><mi>I</mi></mrow></math></span> of positive integers, let <span><math><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>R</mi><mi>S</mi><mi>I</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> be the number of partitions of <span><math><mi>n</mi></math></span> with parts from <span><math><mrow><mi>R</mi><mo>∪</mo><mi>S</mi><mo>∪</mo><mi>I</mi></mrow></math></span> and <span><math><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>R</mi><mo>&gt;</mo><mi>S</mi><mo>,</mo><mi>I</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> be the number of such partitions with a greater number of parts in <span><math><mi>R</mi></math></span> than that in <span><math><mi>S</mi></math></span>. In this paper, in the case that <span><math><mrow><mi>R</mi><mo>,</mo><mi>S</mi><mo>,</mo><mi>I</mi></mrow></math></span> are finite, we obtain an explicit formula of the asymptotic ratio of <span><math><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>R</mi><mo>&gt;</mo><mi>S</mi><mo>,</mo><mi>I</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> to <span><math><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>R</mi><mi>S</mi><mi>I</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span>. The key technique for computing this ratio is to estimate a partition number at the volume of a certain polytope. A conjecture is proposed in the case that <span><math><mrow><mi>R</mi><mo>,</mo><mi>S</mi></mrow></math></span> are certain infinite arithmetic progressions.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"131 ","pages":"Article 104245"},"PeriodicalIF":0.9,"publicationDate":"2025-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145118229","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Higher q-continued fractions","authors":"Amanda Burcroff ,&nbsp;Nicholas Ovenhouse ,&nbsp;Ralf Schiffler ,&nbsp;Sylvester W. Zhang","doi":"10.1016/j.ejc.2025.104244","DOIUrl":"10.1016/j.ejc.2025.104244","url":null,"abstract":"<div><div>We introduce a <span><math><mi>q</mi></math></span>-analog of the higher continued fractions introduced by the last three authors in a previous work (together with Gregg Musiker), which are simultaneously a generalization of the <span><math><mi>q</mi></math></span>-rational numbers of Morier-Genoud and Ovsienko. They are defined as ratios of generating functions for <span><math><mi>P</mi></math></span>-partitions on certain posets. We give matrix formulas for computing them, which generalize previous results in the <span><math><mrow><mi>q</mi><mo>=</mo><mn>1</mn></mrow></math></span> case. We also show that certain properties enjoyed by the <span><math><mi>q</mi></math></span>-rationals are also satisfied by our higher versions.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"131 ","pages":"Article 104244"},"PeriodicalIF":0.9,"publicationDate":"2025-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145096502","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A counterexample to the Ross–Yong conjecture for Grothendieck polynomials","authors":"Colleen Robichaux","doi":"10.1016/j.ejc.2025.104241","DOIUrl":"10.1016/j.ejc.2025.104241","url":null,"abstract":"<div><div>We give a minimal counterexample for a conjecture of Ross and Yong (2015) which proposes a K-Kohnert rule for Grothendieck polynomials. We conjecture a revised version of this rule. We then prove both rules hold in the 321-avoiding case.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"131 ","pages":"Article 104241"},"PeriodicalIF":0.9,"publicationDate":"2025-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145096538","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Polynomial expressions for the dimensions of the representations of symmetric groups and restricted standard Young tableaux","authors":"Avichai Cohen,&nbsp;Shaul Zemel","doi":"10.1016/j.ejc.2025.104242","DOIUrl":"10.1016/j.ejc.2025.104242","url":null,"abstract":"<div><div>Given a partition <span><math><mi>λ</mi></math></span> of a number <span><math><mi>k</mi></math></span>, it is known that by adding a long line of length <span><math><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></math></span>, the dimension of the associated representation of <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is an integer-valued polynomial of degree <span><math><mi>k</mi></math></span> in <span><math><mi>n</mi></math></span>. We show that its expansion in the binomial basis is bounded by the length of <span><math><mi>λ</mi></math></span>, and that the resulting coefficient of index <span><math><mi>h</mi></math></span>, with alternating signs, counts the standard Young tableaux of shape <span><math><mi>λ</mi></math></span> in which a given collection of consecutive <span><math><mi>h</mi></math></span> numbers lie in increasing rows. We also construct bijections in order to demonstrate explicitly that this number is indeed independent of the set of consecutive <span><math><mi>h</mi></math></span> numbers used.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"131 ","pages":"Article 104242"},"PeriodicalIF":0.9,"publicationDate":"2025-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145096500","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Saturation results around the Erdős–Szekeres problem","authors":"Gábor Damásdi ,&nbsp;Zichao Dong ,&nbsp;Manfred Scheucher ,&nbsp;Ji Zeng","doi":"10.1016/j.ejc.2025.104236","DOIUrl":"10.1016/j.ejc.2025.104236","url":null,"abstract":"<div><div>In this paper, we consider saturation problems related to the celebrated Erdős–Szekeres convex polygon problem. For each <span><math><mrow><mi>n</mi><mo>≥</mo><mn>7</mn></mrow></math></span>, we construct a planar point set of size <span><math><mrow><mrow><mo>(</mo><mn>7</mn><mo>/</mo><mn>8</mn><mo>)</mo></mrow><mi>⋅</mi><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msup></mrow></math></span> which is saturated for convex <span><math><mi>n</mi></math></span>-gons. That is, the set contains no <span><math><mi>n</mi></math></span> points in convex position while the addition of any new point creates such a configuration. This demonstrates that the saturation number is smaller than the Ramsey number for the Erdős–Szekeres problem. The proof also shows that the original Erdős–Szekeres construction is indeed saturated. Our construction is based on a similar improvement for the saturation version of the cups-versus-caps theorem. Moreover, we consider the generalization of the cups-versus-caps theorem to monotone paths in ordered hypergraphs. In contrast to the geometric setting, we show that this abstract saturation number is always equal to the corresponding Ramsey number.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"131 ","pages":"Article 104236"},"PeriodicalIF":0.9,"publicationDate":"2025-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145096501","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}