{"title":"Binary sequences meet the Fibonacci sequence","authors":"Piotr Miska , Bartosz Sobolewski , Maciej Ulas","doi":"10.1016/j.aam.2025.102914","DOIUrl":"10.1016/j.aam.2025.102914","url":null,"abstract":"<div><div>We introduce a new family of number sequences <span><math><msub><mrow><mo>(</mo><mi>f</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>)</mo></mrow><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span>, governed by the recurrence relation<span><span><span><math><mi>f</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mi>a</mi><mi>f</mi><mo>(</mo><mi>n</mi><mo>−</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mi>b</mi><mi>f</mi><mo>(</mo><mi>n</mi><mo>−</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>−</mo><mn>2</mn><mo>)</mo><mo>,</mo></math></span></span></span> where <span><math><mi>u</mi><mo>=</mo><msub><mrow><mo>(</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span> is a sequence with values <span><math><mn>0</mn><mo>,</mo><mn>1</mn></math></span>. Our study focuses on the properties of the sequence of quotients <span><math><mi>h</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><mi>f</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> and its set of values <span><math><mi>V</mi><mo>(</mo><mi>f</mi><mo>)</mo><mo>=</mo><mo>{</mo><mi>h</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>:</mo><mi>n</mi><mo>∈</mo><mi>N</mi><mo>}</mo></math></span> for various <strong>u</strong>. We give a sufficient condition for finiteness of <span><math><mi>V</mi><mo>(</mo><mi>f</mi><mo>)</mo></math></span> and automaticity of <span><math><msub><mrow><mo>(</mo><mi>h</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>)</mo></mrow><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span>, which holds in particular when <strong>u</strong> is the famous Prouhet-Thue-Morse sequence. In the automatic case, a constructive approach is used, with the help of the software <span>Walnut</span>. On the other hand, we prove that the set <span><math><mi>V</mi><mo>(</mo><mi>f</mi><mo>)</mo></math></span> is infinite for other special binary sequences <strong>u</strong>, and obtain a trichotomy in its topological type when <strong>u</strong> is eventually periodic.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"169 ","pages":"Article 102914"},"PeriodicalIF":1.0,"publicationDate":"2025-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144169097","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":"Depth and regularity of tableau ideals","authors":"Do Trong Hoang , Thanh Vu","doi":"10.1016/j.aam.2025.102913","DOIUrl":"10.1016/j.aam.2025.102913","url":null,"abstract":"<div><div>We compute the depth and regularity of ideals associated with arbitrary fillings of positive integers to a Young diagram, called the tableau ideals.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"169 ","pages":"Article 102913"},"PeriodicalIF":1.0,"publicationDate":"2025-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144138028","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":"q-Super Catalan numbers: Combinatorial identities, generating functions, and Narayana refinements","authors":"Arthur Rodelet–Causse , Lenny Tevlin","doi":"10.1016/j.aam.2025.102911","DOIUrl":"10.1016/j.aam.2025.102911","url":null,"abstract":"<div><div>We derive a number of combinatorial identities satisfied by the <em>q</em>-super Catalan numbers. In particular, we extend some of the known combinatorial identities (Touchard, Koshy, Reed Dawson) to the <em>q</em>-super Catalan numbers.</div><div>Next, we introduce some <em>q</em>-convolution identities involving q-central binomial and q-Catalan numbers, and derive a generating function for <em>q</em>-Catalan numbers.</div><div>Then we introduce Narayana-type refinements of the super Catalan numbers. We prove algebraically the <em>γ</em>-positivity of those refinements and give a combinatorial proof in a special case through the type B analog of noncrossing partitions. Then we introduce their natural <em>q</em>-analogs, prove their <em>q</em>-<em>γ</em>-positivity, and prove some identities they satisfy, generalizing identities of Kreweras <span><span>[17]</span></span> and Le Jen-Shoo <span><span>[11]</span></span>. Using yet another identity, we prove that these refinements are positive integer polynomials in <em>q</em>.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"169 ","pages":"Article 102911"},"PeriodicalIF":1.0,"publicationDate":"2025-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144116089","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":"Equivalence classes of lower and upper descent weak Bruhat intervals","authors":"Seung-Il Choi , Sun-Young Nam , Young-Tak Oh","doi":"10.1016/j.aam.2025.102910","DOIUrl":"10.1016/j.aam.2025.102910","url":null,"abstract":"<div><div>Let <span><math><mrow><mi>Int</mi></mrow><mo>(</mo><mi>n</mi><mo>)</mo></math></span> denote the set of nonempty left weak Bruhat intervals in the symmetric group <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. We investigate the equivalence relation <figure><img></figure> on <span><math><mrow><mi>Int</mi></mrow><mo>(</mo><mi>n</mi><mo>)</mo></math></span>, where <figure><img></figure> if and only if there exists a descent-preserving poset isomorphism between <em>I</em> and <em>J</em>. For each equivalence class <em>C</em> of <figure><img></figure>, a partial order ⪯ is defined by <span><math><msub><mrow><mo>[</mo><mi>σ</mi><mo>,</mo><mi>ρ</mi><mo>]</mo></mrow><mrow><mi>L</mi></mrow></msub><mo>⪯</mo><msub><mrow><mo>[</mo><msup><mrow><mi>σ</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>,</mo><msup><mrow><mi>ρ</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>]</mo></mrow><mrow><mi>L</mi></mrow></msub></math></span> if and only if <span><math><mi>σ</mi><msub><mrow><mo>⪯</mo></mrow><mrow><mi>R</mi></mrow></msub><msup><mrow><mi>σ</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span>. Kim–Lee–Oh (2024) showed that the poset <span><math><mo>(</mo><mi>C</mi><mo>,</mo><mo>⪯</mo><mo>)</mo></math></span> is isomorphic to a right weak Bruhat interval.</div><div>In this paper, we focus on lower and upper descent weak Bruhat intervals, specifically those of the form <span><math><msub><mrow><mo>[</mo><msub><mrow><mi>w</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>S</mi><mo>)</mo><mo>,</mo><mi>σ</mi><mo>]</mo></mrow><mrow><mi>L</mi></mrow></msub></math></span> or <span><math><msub><mrow><mo>[</mo><mi>σ</mi><mo>,</mo><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>S</mi><mo>)</mo><mo>]</mo></mrow><mrow><mi>L</mi></mrow></msub></math></span>, where <span><math><msub><mrow><mi>w</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>S</mi><mo>)</mo></math></span> is the longest element in the parabolic subgroup <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>S</mi></mrow></msub></math></span> of <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, generated by <span><math><mo>{</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo><mi>i</mi><mo>∈</mo><mi>S</mi><mo>}</mo></math></span> for a subset <span><math><mi>S</mi><mo>⊆</mo><mo>[</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>]</mo></math></span>, and <span><math><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>S</mi><mo>)</mo></math></span> is the longest element among the minimal-length representatives of left <span><math><msub><mrow><mi>S</mi></mrow><mrow><mo>[</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>]</mo><mo>∖</mo><mi>S</mi></mrow></msub></math></span>-cosets in <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. We begin by providing a poset-theoretic characterization of the equivalence relation <figure><img></figure>. Using","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"169 ","pages":"Article 102910"},"PeriodicalIF":1.0,"publicationDate":"2025-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144088875","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 result for hemi-bundled cross-intersecting families","authors":"Yongjiang Wu, Lihua Feng, Yongtao Li","doi":"10.1016/j.aam.2025.102912","DOIUrl":"10.1016/j.aam.2025.102912","url":null,"abstract":"<div><div>Two families <span><math><mi>F</mi></math></span> and <span><math><mi>G</mi></math></span> are called cross-intersecting if for every <span><math><mi>F</mi><mo>∈</mo><mi>F</mi></math></span> and <span><math><mi>G</mi><mo>∈</mo><mi>G</mi></math></span>, the intersection <span><math><mi>F</mi><mo>∩</mo><mi>G</mi></math></span> is non-empty. It is significant to determine the maximum sum of sizes of cross-intersecting families under the additional assumption that one of the two families is intersecting. Such a pair of families is said to be hemi-bundled. In particular, Frankl (2016) proved that for <span><math><mi>k</mi><mo>≥</mo><mn>1</mn><mo>,</mo><mi>t</mi><mo>≥</mo><mn>0</mn></math></span> and <span><math><mi>n</mi><mo>≥</mo><mn>2</mn><mi>k</mi><mo>+</mo><mi>t</mi></math></span>, if <span><math><mi>F</mi><mo>⊆</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mrow><mi>k</mi><mo>+</mo><mi>t</mi></mrow></mtd></mtr></mtable><mo>)</mo></mrow></math></span> and <span><math><mi>G</mi><mo>⊆</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow></math></span> are cross-intersecting families, in which <span><math><mi>F</mi></math></span> is non-empty and <span><math><mo>(</mo><mi>t</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-intersecting, then <span><math><mo>|</mo><mi>F</mi><mo>|</mo><mo>+</mo><mo>|</mo><mi>G</mi><mo>|</mo><mo>≤</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>−</mo><mi>k</mi><mo>−</mo><mi>t</mi></mrow></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow><mo>+</mo><mn>1</mn></math></span>. This bound is achieved when <span><math><mi>F</mi></math></span> consists of a single set. In this paper, we generalize this result under the constraint <span><math><mo>|</mo><mi>F</mi><mo>|</mo><mo>≥</mo><mi>r</mi></math></span> for every <span><math><mi>r</mi><mo>≤</mo><mi>n</mi><mo>−</mo><mi>k</mi><mo>−</mo><mi>t</mi><mo>+</mo><mn>1</mn></math></span>. Moreover, we investigate the stability results of Katona's theorem for non-uniform families with the <em>s</em>-union property. Our result extends the stabilities established by Frankl (2017) and Li and Wu (2024). As applications, we revisit a recent result of Frankl and Wang (2024) as well as a result of Kupavskii (2018). Furthermore, we determine the extremal families in these two results.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"169 ","pages":"Article 102912"},"PeriodicalIF":1.0,"publicationDate":"2025-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144084372","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":"Dissections of lacunary eta quotients and identically vanishing coefficients","authors":"Tim Huber , James McLaughlin , Dongxi Ye","doi":"10.1016/j.aam.2025.102902","DOIUrl":"10.1016/j.aam.2025.102902","url":null,"abstract":"<div><div>For any function <span><math><mi>A</mi><mo>(</mo><mi>q</mi><mo>)</mo><mo>=</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mrow><mo>∞</mo></mrow></msubsup><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> define<span><span><span><math><msub><mrow><mi>A</mi></mrow><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></msub><mo>:</mo><mo>=</mo><mo>{</mo><mi>n</mi><mo>∈</mo><mi>N</mi><mo>:</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mn>0</mn><mo>}</mo><mo>.</mo></math></span></span></span> Now suppose <span><math><mi>C</mi><mo>(</mo><mi>q</mi><mo>)</mo></math></span> and <span><math><mi>D</mi><mo>(</mo><mi>q</mi><mo>)</mo></math></span> are two functions whose <em>m</em>-dissections are given by<span><span><span><math><mi>C</mi><mo>(</mo><mi>q</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>0</mn></mrow></msub><msub><mrow><mi>G</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>)</mo><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>q</mi><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>)</mo><mo>+</mo><mo>…</mo><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msub><msup><mrow><mi>q</mi></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msup><msub><mrow><mi>G</mi></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>)</mo><mo>,</mo></math></span></span></span><span><span><span><math><mi>D</mi><mo>(</mo><mi>q</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>d</mi></mrow><mrow><mn>0</mn></mrow></msub><msub><mrow><mi>G</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>)</mo><mo>+</mo><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>q</mi><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>)</mo><mo>+</mo><mo>…</mo><mo>+</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msub><msup><mrow><mi>q</mi></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msup><msub><mrow><mi>G</mi></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>)</mo><mo>.</mo></math></span></span></span> If it is the case that <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mn>0</mn><mo>⟺</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mn>0</mn></math></span>, <span><math><mi>i</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>m</mi><mo>−</mo><mn>1</mn></math></span>, then we say that <span><math><mi>C</mi><mo>(</mo><mi","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"168 ","pages":"Article 102902"},"PeriodicalIF":1.0,"publicationDate":"2025-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143888196","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":"Influence of the automorphism group of a graph on its PageRank scores of vertices","authors":"Dein Wong , Qi Zhou , Xinlei Wang","doi":"10.1016/j.aam.2025.102900","DOIUrl":"10.1016/j.aam.2025.102900","url":null,"abstract":"<div><div>Google's success derives in large part from its PageRank algorithm, which assign a score to every web page according to its importance. Recently, G. Modjtaba et al. (2021) <span><span>[19]</span></span> proved that similar vertices in a graph have the same PageRank score and they proposed a conjecture, suspecting that two graphs are completely non-Co-PR if they are non-Co-PR graphs. The investigation of this paper mainly concerns the influence of the automorphism group of a graph on its PageRank scores of vertices. The main results of this article are as follows.<ul><li><span>1.</span><span><div>Based on matrix analysis, two conditions on what kinds of vertices have the same PageRank score are obtained.</div></span></li><li><span>2.</span><span><div>Four techniques for constructing Co-PR graphs are established.</div></span></li><li><span>3.</span><span><div>A non-regular connected graph of order <em>n</em>, with <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mi>n</mi></mrow></mfrac></math></span> as PR scores of most of its vertices, is constructed, which provides a negative answer to Modjtaba's conjecture above.</div></span></li></ul></div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"168 ","pages":"Article 102900"},"PeriodicalIF":1.0,"publicationDate":"2025-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143855383","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":"Counting flows of b-compatible graphs","authors":"Houshan Fu , Xiangyu Ren , Suijie Wang","doi":"10.1016/j.aam.2025.102901","DOIUrl":"10.1016/j.aam.2025.102901","url":null,"abstract":"<div><div>Kochol introduced the assigning polynomial <span><math><mi>F</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>α</mi><mo>;</mo><mi>k</mi><mo>)</mo></math></span> to count nowhere-zero <span><math><mo>(</mo><mi>A</mi><mo>,</mo><mi>b</mi><mo>)</mo></math></span>-flows of a graph <em>G</em>, where <em>A</em> is a finite Abelian group and <em>α</em> is a <span><math><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span>-assigning from a family <span><math><mi>Λ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of certain nonempty vertex subsets of <em>G</em> to <span><math><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span>. We introduce the concepts of <em>b</em>-compatible graph and <em>b</em>-compatible broken bond to give an explicit formula for the assigning polynomials and to examine their coefficients. More specifically, for a function <span><math><mi>b</mi><mo>:</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>→</mo><mi>A</mi></math></span>, let <span><math><msub><mrow><mi>α</mi></mrow><mrow><mi>G</mi><mo>,</mo><mi>b</mi></mrow></msub></math></span> be a <span><math><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span>-assigning of <em>G</em> such that for each <span><math><mi>X</mi><mo>∈</mo><mi>Λ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, <span><math><msub><mrow><mi>α</mi></mrow><mrow><mi>G</mi><mo>,</mo><mi>b</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span> if and only if <span><math><msub><mrow><mo>∑</mo></mrow><mrow><mi>v</mi><mo>∈</mo><mi>X</mi></mrow></msub><mi>b</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span>. We show that for any <span><math><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span>-assigning <em>α</em> of <em>G</em>, if there exists a function <span><math><mi>b</mi><mo>:</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>→</mo><mi>A</mi></math></span> such that <em>G</em> is <em>b</em>-compatible and <span><math><mi>α</mi><mo>=</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>G</mi><mo>,</mo><mi>b</mi></mrow></msub></math></span>, then the assigning polynomial <span><math><mi>F</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>α</mi><mo>;</mo><mi>k</mi><mo>)</mo></math></span> has the <em>b</em>-compatible spanning subgraph expansion<span><span><span><math><mi>F</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>α</mi><mo>;</mo><mi>k</mi><mo>)</mo><mo>=</mo><munder><mo>∑</mo><mrow><mtable><mtr><mtd><mi>S</mi><mo>⊆</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>,</mo></mtd></mtr><mtr><mtd><mi>G</mi><mo>−</mo><mi>S</mi><mrow><mtext> is</mtext><mspace></mspace><mtext>b</mtext><mtext>-compatible</mtext></mrow></mtd></mtr></mtable></mrow></munder><msup><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>|</mo><mi>S</mi><mo>|</mo></mrow></msup><msup><mrow><mi>k</mi></mrow><mrow><mi>m</mi><mo>(</mo><mi>G</mi><mo>−</mo><mi>S</mi><mo>)</mo></mrow></msup><mo>,</mo></math></span></span></span> and is the following form<span><span><span><math><mi>F</mi>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"168 ","pages":"Article 102901"},"PeriodicalIF":1.0,"publicationDate":"2025-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143826047","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":"The maximum number of cycles in a triangular-grid billiards system with a given perimeter","authors":"Honglin Zhu","doi":"10.1016/j.aam.2025.102888","DOIUrl":"10.1016/j.aam.2025.102888","url":null,"abstract":"<div><div>Given a grid polygon <em>P</em> in a grid of equilateral triangles, Defant and Jiradilok considered a billiards system where beams of light bounce around inside <em>P</em>. We study the relationship between the perimeter <span><math><mi>perim</mi><mo>(</mo><mi>P</mi><mo>)</mo></math></span> of <em>P</em> and the number of different trajectories <span><math><mi>cyc</mi><mo>(</mo><mi>P</mi><mo>)</mo></math></span> that the billiards system has. Resolving a conjecture of Defant and Jiradilok, we prove the sharp inequality <span><math><mi>cyc</mi><mo>(</mo><mi>P</mi><mo>)</mo><mo>≤</mo><mo>(</mo><mi>perim</mi><mo>(</mo><mi>P</mi><mo>)</mo><mo>+</mo><mn>2</mn><mo>)</mo><mo>/</mo><mn>4</mn></math></span> and characterize the equality cases.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"168 ","pages":"Article 102888"},"PeriodicalIF":1.0,"publicationDate":"2025-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143826046","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":"Colored q-Stirling and q-Lah numbers: A new view continued","authors":"Sen-Peng Eu , Louis Kao , Juei-Yin Lin","doi":"10.1016/j.aam.2025.102889","DOIUrl":"10.1016/j.aam.2025.102889","url":null,"abstract":"<div><div>Cai and Readdy proposed a new framework for studying the <em>q</em>-analogue <span><math><mi>f</mi><mo>(</mo><mi>q</mi><mo>)</mo></math></span> of a combinatorial structure <em>S</em>. Specifically, the aim is to identify two statistics over <em>S</em> and a proper subset <span><math><msup><mrow><mi>S</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> of <em>S</em> such that <span><math><mi>f</mi><mo>(</mo><mi>q</mi><mo>)</mo></math></span> represents the <em>q</em>-<span><math><mo>(</mo><mn>1</mn><mo>+</mo><mi>q</mi><mo>)</mo></math></span>-expansion over <span><math><msup><mrow><mi>S</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span>, and to explore the poset and topological interpretations of this expansion. Cai and Readdy provided comprehensive profiles for classical Stirling numbers of both kinds within this framework. In this work, we extend Cai and Readdy's results to colored <em>q</em>-Stirling numbers of both kinds, as well as colored <em>q</em>-Lah numbers. We also briefly discuss <em>q</em>-Stirling and <em>q</em>-Lah numbers of type <em>D</em>.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"168 ","pages":"Article 102889"},"PeriodicalIF":1.0,"publicationDate":"2025-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143808186","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}
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