Pub Date : 2025-06-18DOI: 10.1016/j.aam.2025.102926
Jesse Campion Loth, Amarpreet Rattan
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 α and β, the average number of cycles in their product is between and , where is the harmonic number.
{"title":"On the average number of cycles in conjugacy class products","authors":"Jesse Campion Loth, 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":"2025-06-18","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}
Pub Date : 2025-06-09DOI: 10.1016/j.aam.2025.102916
Yaroslav Shitov
The positive part of the field consists of Puiseux series with positive real leading terms. Answering a question of Yu, we show that, if M is an matrix with entries in and rank two, then there are an matrix A and matrix B with entries in such that . We discuss the problem in larger ranks and answer a further question arisen in a work of Brandenburg, Loho, and Sinn.
{"title":"On the nonnegative ranks of matrices in Puiseux series fields","authors":"Yaroslav Shitov","doi":"10.1016/j.aam.2025.102916","DOIUrl":"10.1016/j.aam.2025.102916","url":null,"abstract":"<div><div>The <em>positive part</em> <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span> of the field <span><math><mi>C</mi><mo>{</mo><mo>{</mo><mi>t</mi><mo>}</mo><mo>}</mo></math></span> consists of Puiseux series with positive real leading terms. Answering a question of Yu, we show that, if <em>M</em> is an <span><math><mi>m</mi><mo>×</mo><mi>n</mi></math></span> matrix with entries in <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span> and rank two, then there are an <span><math><mi>m</mi><mo>×</mo><mn>2</mn></math></span> matrix <em>A</em> and <span><math><mn>2</mn><mo>×</mo><mi>n</mi></math></span> matrix <em>B</em> with entries in <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span> such that <span><math><mi>M</mi><mo>=</mo><mi>A</mi><mi>B</mi></math></span>. We discuss the problem in larger ranks and answer a further question arisen in a work of Brandenburg, Loho, and Sinn.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"170 ","pages":"Article 102916"},"PeriodicalIF":1.0,"publicationDate":"2025-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144239672","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}
Pub Date : 2025-06-02DOI: 10.1016/j.aam.2025.102915
Jianfeng Wang , Jing Wang , Maurizio Brunetti , Francesco Belardo , Ligong Wang
For each squared graph matrix M, the Hoffman program consists of two aspects: finding all the possible limit points of M-spectral radii of graphs and detecting all the connected graphs whose M-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 -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 .
{"title":"Developments on the Hoffman program of graphs","authors":"Jianfeng Wang , Jing Wang , Maurizio Brunetti , Francesco Belardo , 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":"2025-06-02","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}
Pub Date : 2025-05-29DOI: 10.1016/j.aam.2025.102914
Piotr Miska , Bartosz Sobolewski , Maciej Ulas
We introduce a new family of number sequences , governed by the recurrence relation where is a sequence with values . Our study focuses on the properties of the sequence of quotients and its set of values for various u. We give a sufficient condition for finiteness of and automaticity of , which holds in particular when u is the famous Prouhet-Thue-Morse sequence. In the automatic case, a constructive approach is used, with the help of the software Walnut. On the other hand, we prove that the set is infinite for other special binary sequences u, and obtain a trichotomy in its topological type when u is eventually periodic.
{"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}
Pub Date : 2025-05-26DOI: 10.1016/j.aam.2025.102913
Do Trong Hoang , Thanh Vu
We compute the depth and regularity of ideals associated with arbitrary fillings of positive integers to a Young diagram, called the tableau ideals.
我们计算了杨氏图中任意正整数填充的理想的深度和规律性，称为表理想。
{"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}
Pub Date : 2025-05-23DOI: 10.1016/j.aam.2025.102911
Arthur Rodelet–Causse , Lenny Tevlin
We derive a number of combinatorial identities satisfied by the q-super Catalan numbers. In particular, we extend some of the known combinatorial identities (Touchard, Koshy, Reed Dawson) to the q-super Catalan numbers.
Next, we introduce some q-convolution identities involving q-central binomial and q-Catalan numbers, and derive a generating function for q-Catalan numbers.
Then we introduce Narayana-type refinements of the super Catalan numbers. We prove algebraically the γ-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 q-analogs, prove their q-γ-positivity, and prove some identities they satisfy, generalizing identities of Kreweras [17] and Le Jen-Shoo [11]. Using yet another identity, we prove that these refinements are positive integer polynomials in q.
我们得到了一些由q-超加泰罗尼亚数满足的组合恒等式。特别地，我们将一些已知的组合恒等式（Touchard, Koshy, Reed Dawson）推广到q-超加泰罗尼亚数。其次，我们引入了一些涉及q-中心二项式和q-加泰罗尼亚数的q-卷积恒等式，并推导了q-加泰罗尼亚数的生成函数。然后我们引入了超加泰罗尼亚数的narayana型细化。我们从代数上证明了这些改进的γ-正性，并通过非交叉分区的B型模拟给出了一个特殊情况下的组合证明。然后引入它们的天然q-类似物，证明了它们的q-γ-正性，并证明了它们满足的一些恒等式，推广了Kreweras[17]和Le jen - sho[11]的恒等式。利用另一个恒等式，我们证明了这些改进是q的正整数多项式。
{"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}
Pub Date : 2025-05-20DOI: 10.1016/j.aam.2025.102910
Seung-Il Choi , Sun-Young Nam , Young-Tak Oh
<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
{"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}
Pub Date : 2025-05-19DOI: 10.1016/j.aam.2025.102912
Yongjiang Wu, Lihua Feng, Yongtao Li
Two families and are called cross-intersecting if for every and , the intersection 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 and , if and are cross-intersecting families, in which is non-empty and -intersecting, then . This bound is achieved when consists of a single set. In this paper, we generalize this result under the constraint for every . Moreover, we investigate the stability results of Katona's theorem for non-uniform families with the s-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.
{"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}
Pub Date : 2025-04-22DOI: 10.1016/j.aam.2025.102900
Dein Wong , Qi Zhou , Xinlei Wang
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) [19] 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.
1.
Based on matrix analysis, two conditions on what kinds of vertices have the same PageRank score are obtained.
2.
Four techniques for constructing Co-PR graphs are established.
3.
A non-regular connected graph of order n, with as PR scores of most of its vertices, is constructed, which provides a negative answer to Modjtaba's conjecture above.
{"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}
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