Advances in Applied Mathematics最新文献

{"title":"Polynomial perturbations of Euler's and Clausen's identities","authors":"Dmitrii Karp","doi":"10.1016/j.aam.2026.103042","DOIUrl":"10.1016/j.aam.2026.103042","url":null,"abstract":"<div><div>A product of two hypergeometric series is generally not hypergeometric. However, there are a few cases when such a product does reduce to a single hypergeometric series. The oldest result of this type, beyond the obvious <span><math><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>x</mi><mo>)</mo></mrow><mrow><mi>a</mi></mrow></msup><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>x</mi><mo>)</mo></mrow><mrow><mi>b</mi></mrow></msup><mo>=</mo><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>x</mi><mo>)</mo></mrow><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow></msup></math></span>, is Euler's transformation for the Gauss hypergeometric function <span><math><mmultiscripts><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow><none></none><mprescripts></mprescripts><mrow><mn>2</mn></mrow><none></none></mmultiscripts></math></span>. Another important one is the celebrated Clausen's identity, dating back to 1828, which expresses the square of a suitable <span><math><mmultiscripts><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow><none></none><mprescripts></mprescripts><mrow><mn>2</mn></mrow><none></none></mmultiscripts></math></span> function as a single <span><math><mmultiscripts><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow><none></none><mprescripts></mprescripts><mrow><mn>3</mn></mrow><none></none></mmultiscripts></math></span>. By equating coefficients, each product identity corresponds to a special type of summation theorem for terminating series. Over the last two decades Euler's transformations and many summation theorems have been extended by introducing additional parameter pairs differing by positive integers. This amounts to multiplication of the power series coefficients by values of a fixed polynomial at nonnegative integers. The main goal of this paper is to present an extension of Clausen's identity obtained by such polynomial perturbation. To this end, we first reconsider the polynomial perturbations of Euler's transformations found by Miller and Paris around 2010. We propose new, simplified proofs of their transformations relating them to polynomial interpolation and exhibiting various new forms of the characteristic polynomials. We further introduce the notion of the Miller-Paris operators which play a prominent role in the construction of the extended Clausen's identity.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"175 ","pages":"Article 103042"},"PeriodicalIF":1.3,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981045","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":"Graphs missing a connected partition","authors":"Foster Tom","doi":"10.1016/j.aam.2026.103044","DOIUrl":"10.1016/j.aam.2026.103044","url":null,"abstract":"<div><div>We prove that a graph with a cut vertex whose deletion produces at least five connected components must be missing a connected partition of some type. We prove that this also holds if there are four connected components that each have at least two vertices. In particular, the chromatic symmetric function of such a graph cannot be <em>e</em>-positive. This brings us very close to the conjecture by Dahlberg, She, and van Willigenburg of non-<em>e</em>-positivity for all trees with a vertex of degree at least four. We also prove that spiders with four legs cannot have an <em>e</em>-positive chromatic symmetric function.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"175 ","pages":"Article 103044"},"PeriodicalIF":1.3,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981044","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 a question about real-rooted polynomials and f-polynomials of simplicial complexes","authors":"Lili Mu ,&nbsp;Volkmar Welker","doi":"10.1016/j.aam.2026.103041","DOIUrl":"10.1016/j.aam.2026.103041","url":null,"abstract":"<div><div>For a polynomial <span><math><mi>f</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mn>1</mn><mo>+</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>0</mn></mrow></msub><mi>t</mi><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msub><msup><mrow><mi>t</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> with positive integer coefficients Bell and Skandera (2007) <span><span>[2]</span></span> ask if real-rootedness of <span><math><mi>f</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span> implies that there is a simplicial complex with <em>f</em>-vector <span><math><mo>(</mo><mn>1</mn><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>)</mo></math></span> or equivalently a simplicial complex with <em>f</em>-polynomial <span><math><mi>f</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span>. In this paper we discover properties implied by the real-rootedness of <span><math><mi>f</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span> in terms of the binomial representation <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mrow><mo>(</mo><mtable><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></mtd></mtr><mtr><mtd><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></mtd></mtr></mtable><mo>)</mo></mrow></math></span>, <span><math><mi>i</mi><mo>≥</mo><mn>0</mn></math></span>. We use these to partially answer the question by Bell and Skandera. We also describe two further approaches to the question and use one to verify that some well studied real-rooted classical polynomials are <em>f</em>-polynomials.</div><div>Finally, we provide a series of results showing that the set of <em>f</em>-vectors of simplicial complexes is closed under constructions also preserving real-rootedness of their generating polynomials.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"175 ","pages":"Article 103041"},"PeriodicalIF":1.3,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981043","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":"Capacity bounds on integral flows and the Kostant partition function","authors":"Jonathan Leake ,&nbsp;Alejandro H. Morales","doi":"10.1016/j.aam.2025.103002","DOIUrl":"10.1016/j.aam.2025.103002","url":null,"abstract":"<div><div>The type <em>A</em> Kostant partition function is an important combinatorial object with various applications: it counts integer flows on the complete directed graph, computes Hilbert series of spaces of diagonal harmonics, and can be used to compute weight and tensor product multiplicities of representations. In this paper we study asymptotics of the Kostant partition function, improving on various previously known lower bounds and settling conjectures of O'Neill and Yip. Our methods build upon recent results and techniques of Brändén-Leake-Pak, who used Lorentzian polynomials and Gurvits' capacity method to bound the number of lattice points of transportation and flow polytopes. Finally, we also give new two-sided bounds using the Lidskii formulas from subdivisions of flow polytopes.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"175 ","pages":"Article 103002"},"PeriodicalIF":1.3,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981042","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 vector bundle approach to Nash equilibria","authors":"Hirotachi Abo ,&nbsp;Irem Portakal ,&nbsp;Luca Sodomaco","doi":"10.1016/j.aam.2025.103028","DOIUrl":"10.1016/j.aam.2025.103028","url":null,"abstract":"<div><div>We use vector bundles to study the locus of totally mixed Nash equilibria of an <em>n</em>-player game in normal form, which we call the Nash equilibrium scheme. When the payoff tensor format is balanced, we study the Nash discriminant variety, i.e., the algebraic variety of games whose Nash equilibrium scheme is nonreduced or has a positive dimensional component. We prove that this variety has codimension one. We classify all possible components of the Nash equilibrium scheme for a binary three-player game. We prove that if the payoff tensor is of boundary format, then the Nash discriminant variety has two components: an irreducible hypersurface and a larger-codimensional component. A generic game with an unbalanced payoff tensor format does not admit totally mixed Nash equilibria. We define the Nash resultant variety of games admitting a positive number of totally mixed Nash equilibria. We prove that it is irreducible and determine its codimension and degree.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"175 ","pages":"Article 103028"},"PeriodicalIF":1.3,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145915244","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":"Mixing times of a Burnside process Markov chain on set partitions","authors":"J.E. Paguyo","doi":"10.1016/j.aam.2026.103047","DOIUrl":"10.1016/j.aam.2026.103047","url":null,"abstract":"<div><div>Let <em>X</em> be a finite set and let <em>G</em> be a finite group acting on <em>X</em>. The group action splits <em>X</em> into disjoint orbits. The Burnside process is a Markov chain on <em>X</em> which has a uniform stationary distribution when the chain is lumped to orbits. We consider the case where <span><math><mi>X</mi><mo>=</mo><msup><mrow><mo>[</mo><mi>k</mi><mo>]</mo></mrow><mrow><mi>n</mi></mrow></msup></math></span> with <span><math><mi>k</mi><mo>≥</mo><mi>n</mi></math></span> and <span><math><mi>G</mi><mo>=</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> is the symmetric group on <span><math><mo>[</mo><mi>k</mi><mo>]</mo></math></span>, such that <em>G</em> acts on <em>X</em> by permuting the value of each coordinate. The resulting Burnside process gives a novel algorithm for sampling a set partition of <span><math><mo>[</mo><mi>n</mi><mo>]</mo></math></span> uniformly at random. We obtain bounds on the mixing time and show that the chain is rapidly mixing. For the case <span><math><mi>k</mi><mo>&lt;</mo><mi>n</mi></math></span>, the algorithm corresponds to sampling a set partition of <span><math><mo>[</mo><mi>n</mi><mo>]</mo></math></span> with at most <em>k</em> blocks, and we obtain a mixing time bound which is independent of <em>n</em>. Along the way, we obtain explicit formulas for the transition probabilities and bounds on the second largest eigenvalue for both the original process and the lumped chain.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"175 ","pages":"Article 103047"},"PeriodicalIF":1.3,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146079390","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":"Round Aztec windows, a dual of the Aztec diamond theorem and a curious symmetry of the correlation of diagonal slits","authors":"Mihai Ciucu","doi":"10.1016/j.aam.2026.103049","DOIUrl":"10.1016/j.aam.2026.103049","url":null,"abstract":"<div><div>Fairly shortly after the publication of the Aztec diamond theorem of Elkies, Kuperberg, Larsen and Propp in 1992, interest arose in finding the number of domino tilings of an Aztec diamond with an “Aztec window,” i.e. a hole in the shape of a smaller Aztec diamond at its center. Several intriguing patterns were discovered for the number of tilings of such regions, but the numbers themselves were not “round” — they didn't seem to be given by a simple product formula. In this paper we consider a very closely related shape of holes (namely, odd Aztec rectangles), and prove that a large variety of regions obtained from Aztec rectangles by making such holes in them possess the sought-after property that the number of their domino tilings is given by a simple product formula. We find the same to be true for certain symmetric cruciform regions. We also consider graphs obtained from a toroidal Aztec diamond by making such holes in them, and prove a simple formula that governs the way the number of their perfect matchings changes under a natural evolution of the holes. This yields in particular a natural dual of the Aztec diamond theorem. Some implications for the correlation of such holes are also presented, including an unexpected symmetry for the correlation of diagonal slits on the square grid.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"175 ","pages":"Article 103049"},"PeriodicalIF":1.3,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039053","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 refinement of the Ewens sampling formula","authors":"Eugene Strahov","doi":"10.1016/j.aam.2026.103043","DOIUrl":"10.1016/j.aam.2026.103043","url":null,"abstract":"<div><div>We consider an infinitely-many neutral allelic model of population genetics where all alleles are divided into a finite number of classes, and each class is characterized by its own mutation rate. For this model the allelic composition of a sample taken from a very large population of genes is characterized by a random matrix, and the problem is to describe the joint distribution of the matrix entries. The answer is given by a new generalization of the classical Ewens sampling formula called the refined Ewens sampling formula in this paper. We discuss a Poisson approximation for the refined Ewens sampling formula and present its derivation by several methods. As an application, we obtain limit theorems for the numbers of alleles in different asymptotic regimes.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"175 ","pages":"Article 103043"},"PeriodicalIF":1.3,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039050","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 quasi-tree expansion for the surface Tutte polynomial","authors":"Maya Thompson","doi":"10.1016/j.aam.2026.103045","DOIUrl":"10.1016/j.aam.2026.103045","url":null,"abstract":"<div><div>The surface Tutte polynomial has recently been generalised to pseudo-surfaces equipping it with recursive deletion-contraction relations <span><span>[15]</span></span>. We use these relations to show that this generalisation naturally possesses a quasi-tree expansion. This extends quasi-tree expansions of the Bollobás–Riordan, Las Vergnas and Krushkal polynomials <span><span>[3]</span></span>, <span><span>[4]</span></span>, <span><span>[18]</span></span>, which we recover from our main result.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"175 ","pages":"Article 103045"},"PeriodicalIF":1.3,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039052","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":"An analogue of a formula of Popov","authors":"Pedro Ribeiro","doi":"10.1016/j.aam.2025.103021","DOIUrl":"10.1016/j.aam.2025.103021","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>r</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> denote the number of representations of the positive integer <em>n</em> as the sum of <em>k</em> squares. We prove a new summation formula involving <span><math><msub><mrow><mi>r</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> and the Bessel functions of the first kind, which constitutes an analogue of a result due to the Russian mathematician A. I. Popov.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"175 ","pages":"Article 103021"},"PeriodicalIF":1.3,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039051","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}