{"title":"Mahonian-Stirling statistics for partial permutations","authors":"Ming-Jian Ding , Jiang Zeng","doi":"10.1016/j.aam.2024.102702","DOIUrl":"https://doi.org/10.1016/j.aam.2024.102702","url":null,"abstract":"<div><p>Recently Cheng et al. (2023) <span>[7]</span> generalized the inversion number to partial permutations, which are also known as Laguerre digraphs, and asked for a suitable analogue of MacMahon's major index. We provide such a major index, namely, the corresponding maj and inv statistics are equidistributed, and exhibit a Haglund-Remmel-Wilson type identity. We then interpret some Jacobi-Rogers polynomials in terms of Laguerre digraphs generalizing Deb and Sokal's alternating Laguerre digraph interpretation of some special Jacobi-Rogers polynomials.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"157 ","pages":"Article 102702"},"PeriodicalIF":1.1,"publicationDate":"2024-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140554139","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":"Extremal problems on planar graphs without k edge-disjoint cycles","authors":"Mingqing Zhai , Muhuo Liu","doi":"10.1016/j.aam.2024.102701","DOIUrl":"https://doi.org/10.1016/j.aam.2024.102701","url":null,"abstract":"<div><p>In the 1960s, Erdős and his cooperators initiated the research of the maximum numbers of edges in a graph or a planar graph on <em>n</em> vertices without <em>k</em> edge-disjoint cycles. This problem had been solved for <span><math><mi>k</mi><mo>≤</mo><mn>4</mn></math></span>. As pointed out by Bollobás, it is very difficult for general <em>k</em>. Recently, Tait and Tobin [J. Comb. Theory, Ser. B, 2017] confirmed a famous conjecture on maximum spectral radius of <em>n</em>-vertex planar graphs. Motivated by the above results, we consider two extremal problems on planar graphs without <em>k</em> edge-disjoint cycles. We first determine the maximum number of edges in a planar graph of order <em>n</em> and maximum degree <span><math><mi>n</mi><mo>−</mo><mn>1</mn></math></span> without <em>k</em> edge-disjoint cycles. Based on this, we then determine the maximum spectral radius as well as its unique extremal graph over all planar graphs on <em>n</em> vertices without <em>k</em> edge-disjoint cycles. Finally, we also discuss several extremal problems for general graphs.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"157 ","pages":"Article 102701"},"PeriodicalIF":1.1,"publicationDate":"2024-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140554138","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":"SL(n) contravariant function-valued valuations on polytopes","authors":"Zhongwen Tang , Jin Li , Gangsong Leng","doi":"10.1016/j.aam.2024.102693","DOIUrl":"https://doi.org/10.1016/j.aam.2024.102693","url":null,"abstract":"<div><p>We present a complete classification of <span><math><mi>SL</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> contravariant, <span><math><mi>C</mi><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>∖</mo><mo>{</mo><mi>o</mi><mo>}</mo><mo>)</mo></math></span>-valued valuations on polytopes, without any additional assumptions. It extends the previous results of the second author Li (2020) <span>[10]</span> which have a good connection with the <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> and Orlicz Brunn-Minkowski theory. Additionally, our results deduce a complete classification of <span><math><mi>SL</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> contravariant symmetric-tensor-valued valuations on polytopes.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"157 ","pages":"Article 102693"},"PeriodicalIF":1.1,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140339250","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}
Jordi Castellví , Michael Drmota , Marc Noy , Clément Requilé
{"title":"Chordal graphs with bounded tree-width","authors":"Jordi Castellví , Michael Drmota , Marc Noy , Clément Requilé","doi":"10.1016/j.aam.2024.102700","DOIUrl":"https://doi.org/10.1016/j.aam.2024.102700","url":null,"abstract":"<div><p>Given <span><math><mi>t</mi><mo>≥</mo><mn>2</mn></math></span> and <span><math><mn>0</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>t</mi></math></span>, we prove that the number of labelled <em>k</em>-connected chordal graphs with <em>n</em> vertices and tree-width at most <em>t</em> is asymptotically <span><math><mi>c</mi><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mn>5</mn><mo>/</mo><mn>2</mn></mrow></msup><msup><mrow><mi>γ</mi></mrow><mrow><mi>n</mi></mrow></msup><mi>n</mi><mo>!</mo></math></span>, as <span><math><mi>n</mi><mo>→</mo><mo>∞</mo></math></span>, for some constants <span><math><mi>c</mi><mo>,</mo><mi>γ</mi><mo>></mo><mn>0</mn></math></span> depending on <em>t</em> and <em>k</em>. Additionally, we show that the number of <em>i</em>-cliques (<span><math><mn>2</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>t</mi></math></span>) in a uniform random <em>k</em>-connected chordal graph with tree-width at most <em>t</em> is normally distributed as <span><math><mi>n</mi><mo>→</mo><mo>∞</mo></math></span>.</p><p>The asymptotic enumeration of graphs of tree-width at most <em>t</em> is wide open for <span><math><mi>t</mi><mo>≥</mo><mn>3</mn></math></span>. To the best of our knowledge, this is the first non-trivial class of graphs with bounded tree-width where the asymptotic counting problem is solved. Our starting point is the work of Wormald (1985) <span>[21]</span>, were an algorithm is developed to obtain the exact number of labelled chordal graphs on <em>n</em> vertices.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"157 ","pages":"Article 102700"},"PeriodicalIF":1.1,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140321271","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":"Convolution formulas for multivariate arithmetic Tutte polynomials","authors":"Tianlong Ma, Xian'an Jin, Weiling Yang","doi":"10.1016/j.aam.2024.102692","DOIUrl":"https://doi.org/10.1016/j.aam.2024.102692","url":null,"abstract":"<div><p>The multivariate arithmetic Tutte polynomial of arithmetic matroids is a generalization of the multivariate Tutte polynomial of matroids. In this note, we give the convolution formulas for the multivariate arithmetic Tutte polynomial of the product of two arithmetic matroids. In particular, the convolution formulas for the multivariate arithmetic Tutte polynomial of an arithmetic matroid are obtained. Applying our results, several known convolution formulas including <span>[5, Theorem 10.9 and Corollary 10.10]</span> and <span>[1, Theorems 1 and 4]</span> are proved by a purely combinatorial proof. The proofs presented here are significantly shorter than the previous ones. In addition, we obtain a convolution formula for the characteristic polynomial of an arithmetic matroid.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"157 ","pages":"Article 102692"},"PeriodicalIF":1.1,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140209511","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":"Complements of Schubert polynomials","authors":"Neil J.Y. Fan , Peter L. Guo , Nicolas Y. Liu","doi":"10.1016/j.aam.2024.102691","DOIUrl":"https://doi.org/10.1016/j.aam.2024.102691","url":null,"abstract":"<div><p>Let <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>w</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span> be the Schubert polynomial for a permutation <em>w</em> of <span><math><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></math></span>. For any given composition <em>μ</em>, we say that <span><math><msup><mrow><mi>x</mi></mrow><mrow><mi>μ</mi></mrow></msup><msub><mrow><mi>S</mi></mrow><mrow><mi>w</mi></mrow></msub><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span> is the complement of <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>w</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span> with respect to <em>μ</em>. When each part of <em>μ</em> is equal to <span><math><mi>n</mi><mo>−</mo><mn>1</mn></math></span>, Huh, Matherne, Mészáros and St. Dizier proved that the normalization of <span><math><msup><mrow><mi>x</mi></mrow><mrow><mi>μ</mi></mrow></msup><msub><mrow><mi>S</mi></mrow><mrow><mi>w</mi></mrow></msub><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span> is a Lorentzian polynomial. They further conjectured that the normalization of <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>w</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span> is Lorentzian. It can be shown that if there exists a composition <em>μ</em> such that <span><math><msup><mrow><mi>x</mi></mrow><mrow><mi>μ</mi></mrow></msup><msub><mrow><mi>S</mi></mrow><mrow><mi>w</mi></mrow></msub><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span> is a Schubert polynomial, then the normalization of <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>w</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span> will be Lorentzian. This motivates us to investigate the problem of when <span><math><msup><mrow><mi>x</mi></mrow><mrow><mi>μ</mi></mrow></msup><msub><mrow><mi>S</mi></mrow><mrow><mi>w</mi></mrow></msub><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span> is a Schubert polynomial. We show that if <span><math><msup><mrow><mi>x</mi></mrow><mrow><mi>μ</mi></mrow></msup><msub><mrow><mi>S</mi></mrow><mrow><mi>w</mi></mrow></msub><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span> is a Schubert polynomial, then <em>μ</em> must be a partition. We also consider the case when <em>μ</em> is the staircase partition <span><math><msub><mrow><mi>δ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>)</mo></math></span>, and obtain that <span><math><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>δ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msup><msub><mrow><mi>S</mi></mrow><mrow><mi>w</mi></mrow></msub><mo>(</mo><msup><mrow><mi>x</mi></mrow><","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"157 ","pages":"Article 102691"},"PeriodicalIF":1.1,"publicationDate":"2024-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140188239","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 card guessing games: Limit law for no feedback one-time riffle shuffle","authors":"Markus Kuba , Alois Panholzer","doi":"10.1016/j.aam.2024.102689","DOIUrl":"https://doi.org/10.1016/j.aam.2024.102689","url":null,"abstract":"<div><p>We consider the following card guessing game with no feedback. An ordered deck of <em>n</em> cards labeled 1 up to <em>n</em> is riffle-shuffled exactly one time. Then, the goal of the game is to maximize the number of correct guesses of the cards. One after another a single card is drawn from the top, the guesser makes a guess without seeing the card and gets no response if the guess was correct or not. Building upon and improving earlier results, we provide a limit law for the number of correct guesses and also show convergence of the integer moments.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"156 ","pages":"Article 102689"},"PeriodicalIF":1.1,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140134017","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":"Adjoints of matroids","authors":"Houshan Fu , Chunming Tang , Suijie Wang","doi":"10.1016/j.aam.2024.102690","DOIUrl":"https://doi.org/10.1016/j.aam.2024.102690","url":null,"abstract":"<div><p>We first show that an adjoint of a loopless matroid is connected if and only if the original matroid is connected. By proving that the opposite lattice of a modular matroid is isomorphic to its extension lattice, we obtain that a modular matroid has only one adjoint (up to isomorphism) which can be given by its opposite lattice. This makes projective geometries become a key ingredient in characterizing the adjoint sequence <span><math><mi>a</mi><msup><mrow><mi>d</mi></mrow><mrow><mn>0</mn></mrow></msup><mi>M</mi><mo>,</mo><mi>a</mi><mi>d</mi><mi>M</mi><mo>,</mo><mi>a</mi><msup><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>M</mi><mo>,</mo><mo>…</mo></math></span> of a connected matroid <em>M</em>. We classify such adjoint sequences into three types: finite, cyclic and convergent. For the first two types, the adjoint sequences eventually stabilize at finite projective geometries except for free matroids. For the last type, the infinite non-repeating adjoint sequences are convergent to infinite projective geometries.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"156 ","pages":"Article 102690"},"PeriodicalIF":1.1,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140134175","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":"Certain extensions of results of Siegel, Wilton and Hardy","authors":"Pedro Ribeiro, Semyon Yakubovich","doi":"10.1016/j.aam.2024.102676","DOIUrl":"https://doi.org/10.1016/j.aam.2024.102676","url":null,"abstract":"<div><p>Recently, Dixit et al. <span>[24]</span> established a very elegant generalization of Hardy's theorem concerning the infinitude of zeros that the Riemann zeta function possesses at its critical line.</p><p>By introducing a general transformation formula for the theta function involving the Bessel and modified Bessel functions of the first kind, we extend their result to a class of Dirichlet series satisfying Hecke's functional equation. In the process, we also find new generalizations of classical identities in Analytic number theory.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"156 ","pages":"Article 102676"},"PeriodicalIF":1.1,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140103337","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}
Dariusz Bugajewski , Dawid Bugajewski , Xiao-Xiong Gan , Piotr Maćkowiak
{"title":"On the recursive and explicit form of the general J.C.P. Miller formula with applications","authors":"Dariusz Bugajewski , Dawid Bugajewski , Xiao-Xiong Gan , Piotr Maćkowiak","doi":"10.1016/j.aam.2024.102688","DOIUrl":"https://doi.org/10.1016/j.aam.2024.102688","url":null,"abstract":"<div><p>The famous J.C.P. Miller formula provides a recurrence algorithm for the composition <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>a</mi></mrow></msub><mo>∘</mo><mi>f</mi></math></span>, where <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>a</mi></mrow></msub></math></span> is the formal binomial series and <em>f</em> is a formal power series, however it requires that <em>f</em> has to be a nonunit.</p><p>In this paper we provide the general J.C.P. Miller formula which eliminates the requirement of nonunitness of <em>f</em> and, instead, we establish a necessary and sufficient condition for the existence of the composition <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>a</mi></mrow></msub><mo>∘</mo><mi>f</mi></math></span>. We also provide the general J.C.P. Miller recurrence algorithm for computing the coefficients of that composition, if <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>a</mi></mrow></msub><mo>∘</mo><mi>f</mi></math></span> is well defined, obviously. Our generalizations cover both the case in which <em>f</em> is a one–variable formal power series and the case in which <em>f</em> is a multivariable formal power series.</p><p>In the central part of this article we state, using some combinatorial techniques, the explicit form of the general J.C.P. Miller formula for one-variable case.</p><p>As applications of these results we provide an explicit formula for the inverses of polynomials and formal power series for which the inverses exist, obviously. We also use our results to investigation of approximate solution to a differential equation which cannot be solved in an explicit way.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"156 ","pages":"Article 102688"},"PeriodicalIF":1.1,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140103338","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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