Advances in Applied Mathematics最新文献

{"title":"The syntactic monoid of θ-free palindromic words","authors":"Kalpana Mahalingam","doi":"10.1016/j.aam.2025.102964","DOIUrl":"10.1016/j.aam.2025.102964","url":null,"abstract":"<div><div>In this paper, we consider words that are free of special type of palindromes, inspired by Watson-Crick complementarity in DNA sequences which we call Watson-Crick palindromic free words or <em>θ</em>-palindromic free words, when <em>θ</em> is an antimorphic involution that incorporates the notion of Watson-Crick complementarity of DNA sequences. We discuss certain algebraic properties of the set of all words that avoids <em>θ</em>-palindromes. We give a complete characterization of the syntactic monoid of the language consisting of all <em>θ</em>-palindromic free words over a given alphabet. We also study the ideals of the corresponding syntactic semigroup obtained using Green's relations. Our results also hold for the more general case, where the Watson-Crick complementarity function is replaced by an arbitrary antimorphic involution.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"172 ","pages":"Article 102964"},"PeriodicalIF":1.3,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145019454","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":"Corrigendum to “Stereologically adapted Crofton formulae for tensor valuations” [Adv. Appl. Math. 160 (2024) 102754]","authors":"Emil Dare","doi":"10.1016/j.aam.2025.102937","DOIUrl":"10.1016/j.aam.2025.102937","url":null,"abstract":"","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"171 ","pages":"Article 102937"},"PeriodicalIF":1.0,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144596798","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":"Fan-complete Ramsey numbers","authors":"Fan Chung ,&nbsp;Qizhong Lin","doi":"10.1016/j.aam.2025.102939","DOIUrl":"10.1016/j.aam.2025.102939","url":null,"abstract":"&lt;div&gt;&lt;div&gt;For graphs &lt;em&gt;G&lt;/em&gt; and &lt;em&gt;H&lt;/em&gt;, we consider Ramsey numbers &lt;span&gt;&lt;math&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; with tight lower bounds, namely, &lt;span&gt;&lt;math&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;χ&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, where &lt;span&gt;&lt;math&gt;&lt;mi&gt;χ&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; denotes the chromatic number of &lt;em&gt;G&lt;/em&gt; and &lt;span&gt;&lt;math&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; denotes the number of vertices in &lt;em&gt;H&lt;/em&gt;. We say &lt;em&gt;H&lt;/em&gt; is &lt;em&gt;G&lt;/em&gt;-good if the equality holds.&lt;/div&gt;&lt;div&gt;Let &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; be the join graph obtained from graphs &lt;em&gt;G&lt;/em&gt; and &lt;em&gt;H&lt;/em&gt; by adding all edges between the disjoint vertex sets of &lt;em&gt;G&lt;/em&gt; and &lt;em&gt;H&lt;/em&gt;. Let &lt;em&gt;nH&lt;/em&gt; denote the union graph of &lt;em&gt;n&lt;/em&gt; disjoint copies of &lt;em&gt;H&lt;/em&gt;. We show that &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;-good if &lt;em&gt;n&lt;/em&gt; is sufficiently large. In particular, the fan-graph &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; is &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;-good if &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, improving previous tower-type lower bounds for &lt;em&gt;n&lt;/em&gt; due to Li and Rousseau (1996). Moreover, we give a stronger lower bound inequality for Ramsey number &lt;span&gt;&lt;math&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; for the case of &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, the complete &lt;em&gt;p&lt;/em&gt;-partite graph with &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;. In particular, using a stability-supersaturation lemma by Fox, He and Wigderson (2023), we show that for any fixed graph &lt;em&gt;H&lt;/em&gt;,&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/m","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"171 ","pages":"Article 102939"},"PeriodicalIF":1.0,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144654595","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 distribution of t-hooks of doubled distinct partitions","authors":"Hyunsoo Cho ,&nbsp;Byungchan Kim ,&nbsp;Eunmi Kim ,&nbsp;Ae Ja Yee","doi":"10.1016/j.aam.2025.102958","DOIUrl":"10.1016/j.aam.2025.102958","url":null,"abstract":"<div><div>Recently, Griffin, Ono, and Tsai examined the distribution of the number of <em>t</em>-hooks in partitions of <em>n</em>, which was later followed by the work of Craig, Ono, and Singh on the distribution of the number of <em>t</em>-hooks in self-conjugate partitions of <em>n</em>. Motivated by these studies, in this paper, we further investigate the number of <em>t</em>-hooks in some subsets of partitions. More specifically, we obtain the generating functions for the number of <em>t</em>-hooks in doubled distinct partitions and the number of <em>t</em>-shifted hooks in strict partitions. Based on these generating functions, we prove that the number of <em>t</em>-hooks in doubled distinct partitions and the number of <em>t</em>-shifted hooks in strict partitions are both asymptotically normally distributed.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"171 ","pages":"Article 102958"},"PeriodicalIF":1.3,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144879033","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 Whitney polynomial for hypermaps","authors":"Robert Cori ,&nbsp;Gábor Hetyei","doi":"10.1016/j.aam.2025.102951","DOIUrl":"10.1016/j.aam.2025.102951","url":null,"abstract":"<div><div>We introduce a Whitney polynomial for hypermaps. For maps, our definition depends only on the underlying graph and coincides with the usual definition, but for general hypermaps it depends on the topological structure. Our invariant satisfies a generalized deletion-contraction recurrence and it may be used to generalize the results of Arratia, Bollobás, Ellis-Monaghan, Martin and Sorkin connecting the circuit partition polynomial to the Martin polynomial of a graph. For hypermaps with hyperedges of length at most three our approach also allows generalizing most results connecting the chromatic polynomial and the flow polynomial with the Tutte polynomial of a graph.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"171 ","pages":"Article 102951"},"PeriodicalIF":1.3,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144780478","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":"Non-commutative friezes and their determinants, the non-commutative Laurent phenomenon for weak friezes, and frieze gluing","authors":"Michael Cuntz ,&nbsp;Thorsten Holm ,&nbsp;Peter Jørgensen","doi":"10.1016/j.aam.2025.102940","DOIUrl":"10.1016/j.aam.2025.102940","url":null,"abstract":"<div><div>This paper studies a non-commutative generalisation of Coxeter friezes due to Berenstein and Retakh. It generalises several earlier results to this situation: A formula for frieze determinants, a <em>T</em>-path formula expressing the Laurent phenomenon, and results on gluing friezes together. One of our tools is a non-commutative version of the weak friezes introduced by Çanakçı and Jørgensen.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"171 ","pages":"Article 102940"},"PeriodicalIF":1.0,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144696533","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":"Lattices in Fq[[T]]d and spiral shifting operators","authors":"Yifeng Huang ,&nbsp;Ruofan Jiang","doi":"10.1016/j.aam.2025.102950","DOIUrl":"10.1016/j.aam.2025.102950","url":null,"abstract":"<div><div>We investigate the algebra and combinatorics of an analogue of the Hermite normal form that classifies finite-index submodules of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><msup><mrow><mo>[</mo><mo>[</mo><mi>T</mi><mo>]</mo><mo>]</mo></mrow><mrow><mi>d</mi></mrow></msup></math></span>. We identity both normal forms as instances of Gröbner basis theory under different monomial orders, where the Hermite normal form corresponds to the lex order, and the new normal form the hlex order. We note that the hlex normal form recovers the Smith normal form, a feature not enjoyed by the Hermite normal form. We also identify the combinatorial structure underlying the cell decomposition induced by the hlex normal form, which appears to be of independent interest. Notably, the statistics tracking the cell dimensions is compatible, in a certain way, with a collection of <em>d</em> “spiral shifting operators” on <span><math><msup><mrow><mi>N</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, which pairwise commute and collectively act freely and transitively. Using these operators, we give direct proofs of some new combinatorial identities obtained by translating the results of Solomon <span><span>[26]</span></span> and Petrogradsky <span><span>[25]</span></span> in terms of the hlex normal form.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"171 ","pages":"Article 102950"},"PeriodicalIF":1.3,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144780617","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":"Cookie cutters: Bisections with fixed shapes","authors":"Patrick Schnider ,&nbsp;Pablo Soberón","doi":"10.1016/j.aam.2025.102957","DOIUrl":"10.1016/j.aam.2025.102957","url":null,"abstract":"<div><div>In a mass partition problem, we are interested in finding equitable partitions of smooth measures in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>. In this manuscript, we study the problem of finding simultaneous bisections of measures using scaled copies of a prescribed set <em>K</em>. We distinguish the problem when we are allowed to use scaled and translated copies of <em>K</em> and the problem when we are allowed to use scaled isometric copies of <em>K</em>. These problems have only previously been studied if <em>K</em> is a half-space or a Euclidean ball. We obtain positive results for simultaneous bisection of any <span><math><mi>d</mi><mo>+</mo><mn>1</mn></math></span> masses for star-shaped compact sets <em>K</em> with non-empty interior, where the conditions on the problem depend on the smoothness of the boundary of <em>K</em>. Additional proofs are included for particular instances of <em>K</em>, such as hypercubes and cylinders, answering positively a conjecture of Soberón and Takahashi. The proof methods are topological and involve new Borsuk–Ulam-type theorems.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"171 ","pages":"Article 102957"},"PeriodicalIF":1.3,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144830959","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":"Proof of a K-theoretic polynomial conjecture of Monical, Pechenik, and Searles","authors":"Laura Pierson","doi":"10.1016/j.aam.2025.102959","DOIUrl":"10.1016/j.aam.2025.102959","url":null,"abstract":"&lt;div&gt;&lt;div&gt;As part of a program to develop &lt;em&gt;K&lt;/em&gt;-theoretic analogues of combinatorially important polynomials, Monical, Pechenik, and Searles (2021) proved two expansion formulas &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;‾&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;β&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;‾&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;‾&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;β&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;‾&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, where each of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;‾&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;‾&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;‾&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;‾&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; is a family of polynomials that forms a basis for &lt;span&gt;&lt;math&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;β&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; indexed by weak compositions &lt;em&gt;a&lt;/em&gt;, and &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;β&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;β&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; are monomials in &lt;em&gt;β&lt;/em&gt; for each pair &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; of weak compositions. The polynomials &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;‾&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; are the &lt;em&gt;&lt;strong&gt;Lascoux atoms&lt;/strong&gt;&lt;/em&gt;, &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;‾&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; are the &lt;em&gt;&lt;strong&gt;kaons&lt;/strong&gt;&lt;/em&gt;, &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;‾&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; are the &lt;em&gt;&lt;strong&gt;quasiLascoux polynomials&lt;/strong&gt;&lt;/em&gt;, and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;‾&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; are the &lt;em&gt;&lt;strong&gt;glide polynomials&lt;/strong&gt;&lt;/e","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"171 ","pages":"Article 102959"},"PeriodicalIF":1.3,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144864296","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 relationship for LYM inequalities between Boolean lattices and linear lattices with applications","authors":"Jiuqiang Liu ,&nbsp;Guihai Yu","doi":"10.1016/j.aam.2025.102935","DOIUrl":"10.1016/j.aam.2025.102935","url":null,"abstract":"<div><div>Sperner theory is one of the most important branches in extremal set theory. It has many applications in the field of operation research, computer science, hypergraph theory and so on. The LYM property has become an important tool for studying Sperner property. In this paper, we provide a general relationship for LYM inequalities between Boolean lattices and linear lattices. As applications, we use this relationship to derive generalizations of some well-known theorems on maximum sizes of families containing no copy of certain poset or certain configuration from Boolean lattices to linear lattices, including generalizations of the well-known Kleitman theorem on families containing no <em>s</em> pairwise disjoint members (a non-uniform variant of the famous Erdős matching conjecture) and Johnston-Lu-Milans theorem and Polymath theorem on families containing no <em>d</em>-dimensional Boolean algebras.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"171 ","pages":"Article 102935"},"PeriodicalIF":1.0,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144613816","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}