Advances in Applied Mathematics最新文献

{"title":"On the average number of cycles in conjugacy class products","authors":"Jesse Campion Loth,&nbsp;Amarpreet Rattan","doi":"10.1016/j.aam.2025.102926","DOIUrl":"10.1016/j.aam.2025.102926","url":null,"abstract":"<div><div>We show that for the product of permutations from two fixed point free conjugacy classes, the average number of cycles is always very similar. Specifically, our main result is that for a randomly chosen pair of fixed point free permutations of cycle types <em>α</em> and <em>β</em>, the average number of cycles in their product is between <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>−</mo><mn>3</mn></math></span> and <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>+</mo><mn>1</mn></math></span>, where <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is the harmonic number.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"170 ","pages":"Article 102926"},"PeriodicalIF":1.0,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144314072","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 generalized Dirichlet beta and Riemann zeta functions and Ramanujan-type formulae for beta and zeta values","authors":"S. Yakubovich","doi":"10.1016/j.aam.2025.102934","DOIUrl":"10.1016/j.aam.2025.102934","url":null,"abstract":"<div><div>We define the generalized Dirichlet beta and Riemann zeta functions in terms of the integrals, involving powers of the hyperbolic secant and cosecant functions. The corresponding functional equations are established. Some consequences of the Ramanujan identity for zeta values at odd integers are investigated and new formulae of the Ramanujan type are obtained. These results are achieved, in particular, involving the Kontorovich-Lebedev transform and the corresponding polynomials introduced by the author.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"170 ","pages":"Article 102934"},"PeriodicalIF":1.0,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144579758","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":"Developments on the Hoffman program of graphs","authors":"Jianfeng Wang ,&nbsp;Jing Wang ,&nbsp;Maurizio Brunetti ,&nbsp;Francesco Belardo ,&nbsp;Ligong Wang","doi":"10.1016/j.aam.2025.102915","DOIUrl":"10.1016/j.aam.2025.102915","url":null,"abstract":"<div><div>For each squared graph matrix <em>M</em>, the Hoffman program consists of two aspects: finding all the possible limit points of <em>M</em>-spectral radii of graphs and detecting all the connected graphs whose <em>M</em>-spectral radius does not exceed a fixed limit point. In this survey, we summarize the results on this topic concerning several graph matrices, including the adjacency, the Laplacian, the signless Laplacian, the Hermitian adjacency and the skew-adjacency matrix of graphs. The correspondent problems related to tensors of hypergraphs are also discussed. Moreover, we obtain new results about the Hoffman program with relation to the <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span>-matrix. In particular, we get two generalized versions of it applicable to nonnegative symmetric matrices with fractional elements. We also retrieve the limit points of spectral radii of (signless) Laplacian matrices of graphs less than <span><math><mn>2</mn><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mrow><mo>(</mo><msup><mrow><mo>(</mo><mn>54</mn><mo>−</mo><mn>6</mn><msqrt><mrow><mn>33</mn></mrow></msqrt><mo>)</mo></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup><mo>+</mo><msup><mrow><mo>(</mo><mn>54</mn><mo>+</mo><mn>6</mn><msqrt><mrow><mn>33</mn></mrow></msqrt><mo>)</mo></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup><mo>)</mo></mrow></math></span>.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"169 ","pages":"Article 102915"},"PeriodicalIF":1.0,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144190253","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 ,&nbsp;Sun-Young Nam ,&nbsp;Young-Tak Oh","doi":"10.1016/j.aam.2025.102910","DOIUrl":"10.1016/j.aam.2025.102910","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Let &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;Int&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; denote the set of nonempty left weak Bruhat intervals 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;. We investigate the equivalence relation &lt;figure&gt;&lt;img&gt;&lt;/figure&gt; on &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;Int&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, where &lt;figure&gt;&lt;img&gt;&lt;/figure&gt; if and only if there exists a descent-preserving poset isomorphism between &lt;em&gt;I&lt;/em&gt; and &lt;em&gt;J&lt;/em&gt;. For each equivalence class &lt;em&gt;C&lt;/em&gt; of &lt;figure&gt;&lt;img&gt;&lt;/figure&gt;, a partial order ⪯ is defined by &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;σ&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;L&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;msup&gt;&lt;mrow&gt;&lt;mi&gt;σ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; if and only if &lt;span&gt;&lt;math&gt;&lt;mi&gt;σ&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;⪯&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;σ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;. Kim–Lee–Oh (2024) showed that the poset &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;⪯&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is isomorphic to a right weak Bruhat interval.&lt;/div&gt;&lt;div&gt;In this paper, we focus on lower and upper descent weak Bruhat intervals, specifically those of the form &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;w&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;σ&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; or &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;σ&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;w&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;S&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, where &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;w&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is the longest element in the parabolic subgroup &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;S&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; of &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;, generated by &lt;span&gt;&lt;math&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;s&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;mi&gt;i&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; for a subset &lt;span&gt;&lt;math&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;⊆&lt;/mo&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;/math&gt;&lt;/span&gt;, and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;w&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;S&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is the longest element among the minimal-length representatives of left &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;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;mo&gt;∖&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;-cosets in &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;. We begin by providing a poset-theoretic characterization of the equivalence relation &lt;figure&gt;&lt;img&gt;&lt;/figure&gt;. Using","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"169 ","pages":"Article 102910"},"PeriodicalIF":1.0,"publicationDate":"2025-09-01","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,&nbsp;Lihua Feng,&nbsp;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":"2025-09-01","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":"Depth and regularity of tableau ideals","authors":"Do Trong Hoang ,&nbsp;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":"2025-09-01","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 ,&nbsp;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":"2025-09-01","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":"Binary sequences meet the Fibonacci sequence","authors":"Piotr Miska ,&nbsp;Bartosz Sobolewski ,&nbsp;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":"2025-09-01","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":"The extra basis in noncommuting variables","authors":"Farid Aliniaeifard,&nbsp;Stephanie van Willigenburg","doi":"10.1016/j.aam.2025.102887","DOIUrl":"10.1016/j.aam.2025.102887","url":null,"abstract":"<div><div>We answer a question of Bergeron, Hohlweg, Rosas, and Zabrocki from 2006 to give a combinatorial description for the coproduct of the <em>x</em>-basis in the Hopf algebra of symmetric functions in noncommuting variables, NCSym, which arises in the theory of Grothendieck bialgebras. We achieve this by applying the theory of Hopf monoids and the Fock functor. We also determine combinatorial expansions of this basis in terms of the monomial and power sum symmetric functions in NCSym, and by taking the commutative image of the <em>x</em>-basis we discover a new multiplicative basis for the algebra of symmetric functions.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"168 ","pages":"Article 102887"},"PeriodicalIF":1.0,"publicationDate":"2025-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143767926","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Counting flows of b-compatible graphs","authors":"Houshan Fu ,&nbsp;Xiangyu Ren ,&nbsp;Suijie Wang","doi":"10.1016/j.aam.2025.102901","DOIUrl":"10.1016/j.aam.2025.102901","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Kochol introduced the assigning polynomial &lt;span&gt;&lt;math&gt;&lt;mi&gt;F&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;α&lt;/mi&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; to count nowhere-zero &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;-flows of a graph &lt;em&gt;G&lt;/em&gt;, where &lt;em&gt;A&lt;/em&gt; is a finite Abelian group and &lt;em&gt;α&lt;/em&gt; is a &lt;span&gt;&lt;math&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-assigning from a family &lt;span&gt;&lt;math&gt;&lt;mi&gt;Λ&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; of certain nonempty vertex subsets of &lt;em&gt;G&lt;/em&gt; to &lt;span&gt;&lt;math&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. We introduce the concepts of &lt;em&gt;b&lt;/em&gt;-compatible graph and &lt;em&gt;b&lt;/em&gt;-compatible broken bond to give an explicit formula for the assigning polynomials and to examine their coefficients. More specifically, for a function &lt;span&gt;&lt;math&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mi&gt;V&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;mi&gt;A&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, let &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; be a &lt;span&gt;&lt;math&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-assigning of &lt;em&gt;G&lt;/em&gt; such that for each &lt;span&gt;&lt;math&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;Λ&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;, &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; if and only if &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;. We show that for any &lt;span&gt;&lt;math&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-assigning &lt;em&gt;α&lt;/em&gt; of &lt;em&gt;G&lt;/em&gt;, if there exists a function &lt;span&gt;&lt;math&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mi&gt;V&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;mi&gt;A&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; such that &lt;em&gt;G&lt;/em&gt; is &lt;em&gt;b&lt;/em&gt;-compatible and &lt;span&gt;&lt;math&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, then the assigning polynomial &lt;span&gt;&lt;math&gt;&lt;mi&gt;F&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;α&lt;/mi&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; has the &lt;em&gt;b&lt;/em&gt;-compatible spanning subgraph expansion&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mi&gt;F&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;α&lt;/mi&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;munder&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;mrow&gt;&lt;mtable&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;⊆&lt;/mo&gt;&lt;mi&gt;E&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;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mrow&gt;&lt;mtext&gt; is&lt;/mtext&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mtext&gt;b&lt;/mtext&gt;&lt;mtext&gt;-compatible&lt;/mtext&gt;&lt;/mrow&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;/mrow&gt;&lt;/munder&gt;&lt;msup&gt;&lt;mrow&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;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&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;S&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; and is the following form&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mi&gt;F&lt;/mi&gt;","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"168 ","pages":"Article 102901"},"PeriodicalIF":1.0,"publicationDate":"2025-07-01","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}