Pub Date : 2026-01-28DOI: 10.1016/j.aam.2026.103050
Aaron Robertson
We extend Deuber's theorem on -sets to hold over the multidimensional positive integer lattices. This leads to a multidimensional Rado theorem where we are guaranteed monochromatic multidimensional points in all finite colorings of where the set of coordinates satisfies the given linear Rado system.
{"title":"A multidimensional Rado Theorem","authors":"Aaron Robertson","doi":"10.1016/j.aam.2026.103050","DOIUrl":"10.1016/j.aam.2026.103050","url":null,"abstract":"<div><div>We extend Deuber's theorem on <span><math><mo>(</mo><mi>m</mi><mo>,</mo><mi>p</mi><mo>,</mo><mi>c</mi><mo>)</mo></math></span>-sets to hold over the multidimensional positive integer lattices. This leads to a multidimensional Rado theorem where we are guaranteed monochromatic multidimensional points in all finite colorings of <span><math><msup><mrow><mo>(</mo><msup><mrow><mi>Z</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>)</mo></mrow><mrow><mi>d</mi></mrow></msup></math></span> where the <span><math><msup><mrow><mi>i</mi></mrow><mrow><mi>th</mi></mrow></msup></math></span> set of coordinates satisfies the <span><math><msup><mrow><mi>i</mi></mrow><mrow><mi>th</mi></mrow></msup></math></span> given linear Rado system.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"176 ","pages":"Article 103050"},"PeriodicalIF":1.3,"publicationDate":"2026-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146080817","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 : 2026-01-28DOI: 10.1016/j.aam.2026.103054
Jian Cao
Polynomial expansions of analytic solutions of the heat equation occupy important positions in disciplines such as mathematics and physics [68]. In this paper, we introduce q-3D hypergeometric polynomials and find their corresponding q-heat equations, which were motivated by Ismail and Zhang (2016) [32] and (2017) [33]. We deduce several types of generating functions for q-3D hypergeometric polynomials and Askey–Wilson type integral involving q-3D hypergeometric polynomials by the method of heat equation type q-partial differential equations. In addition, we generalize some results of Ismail and Zhang (2017) [33], Milne (1997) [49] and Jia (2021) [38].
热方程解析解的多项式展开式在数学、物理等学科中占有重要地位[68]。本文引入了由Ismail and Zhang(2016)[32]和（2017）[33]提出的q-3D超几何多项式，并找到了其对应的q-heat方程。利用热方程型q-偏微分方程的方法推导了q-3D超几何多项式的几种生成函数和涉及q-3D超几何多项式的Askey-Wilson型积分。此外，我们还推广了Ismail and Zhang (2017) b[33]、Milne(1997)[49]和Jia(2021)[38]的一些结果。
{"title":"The generalized q-heat equations for q-3D hypergeometric polynomials with applications to generating functions and Askey–Wilson integrals","authors":"Jian Cao","doi":"10.1016/j.aam.2026.103054","DOIUrl":"10.1016/j.aam.2026.103054","url":null,"abstract":"<div><div>Polynomial expansions of analytic solutions of the heat equation occupy important positions in disciplines such as mathematics and physics <span><span>[68]</span></span>. In this paper, we introduce <em>q</em>-3D hypergeometric polynomials and find their corresponding <em>q</em>-heat equations, which were motivated by Ismail and Zhang (2016) <span><span>[32]</span></span> and (2017) <span><span>[33]</span></span>. We deduce several types of generating functions for <em>q</em>-3D hypergeometric polynomials and Askey–Wilson type integral involving <em>q</em>-3D hypergeometric polynomials by the method of heat equation type <em>q</em>-partial differential equations. In addition, we generalize some results of Ismail and Zhang (2017) <span><span>[33]</span></span>, Milne (1997) <span><span>[49]</span></span> and Jia (2021) <span><span>[38]</span></span>.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"176 ","pages":"Article 103054"},"PeriodicalIF":1.3,"publicationDate":"2026-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146080819","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 : 2026-01-27DOI: 10.1016/j.aam.2026.103046
Gianira N. Alfarano , Eimear Byrne
In this paper, we describe properties of the characteristic polynomial of a weighted lattice and show that it has a recursive description, which we use to obtain results on the critical exponent of q-polymatroids. We give a Critical Theorem for representable q-polymatroids and we provide a lower bound on the critical exponent. We show that q-polymatroids arising from certain families of rank-metric codes attain this lower bound.
{"title":"Recursive properties of the characteristic polynomial of weighted lattices","authors":"Gianira N. Alfarano , Eimear Byrne","doi":"10.1016/j.aam.2026.103046","DOIUrl":"10.1016/j.aam.2026.103046","url":null,"abstract":"<div><div>In this paper, we describe properties of the characteristic polynomial of a weighted lattice and show that it has a recursive description, which we use to obtain results on the critical exponent of <em>q</em>-polymatroids. We give a Critical Theorem for representable <em>q</em>-polymatroids and we provide a lower bound on the critical exponent. We show that <em>q</em>-polymatroids arising from certain families of rank-metric codes attain this lower bound.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"176 ","pages":"Article 103046"},"PeriodicalIF":1.3,"publicationDate":"2026-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146080813","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 : 2026-01-27DOI: 10.1016/j.aam.2026.103051
Shane Chern , Lin Jiu , Shuhan Li , Liuquan Wang
It is a standard result that the Hankel determinants for a sequence stay invariant after performing the binomial transform on this sequence. In this work, we extend the scenario to q-binomial transforms and study the behavior of the leading coefficient in such Hankel determinants. We also investigate the leading coefficient in the Hankel determinants for even-indexed Bernoulli polynomials with recourse to a curious binomial transform. In particular, the degrees of these Hankel determinants share the same nature as those in one of the q-binomial cases.
{"title":"Leading coefficient in the Hankel determinants related to binomial and q-binomial transforms","authors":"Shane Chern , Lin Jiu , Shuhan Li , Liuquan Wang","doi":"10.1016/j.aam.2026.103051","DOIUrl":"10.1016/j.aam.2026.103051","url":null,"abstract":"<div><div>It is a standard result that the Hankel determinants for a sequence stay invariant after performing the binomial transform on this sequence. In this work, we extend the scenario to <em>q</em>-binomial transforms and study the behavior of the leading coefficient in such Hankel determinants. We also investigate the leading coefficient in the Hankel determinants for even-indexed Bernoulli polynomials with recourse to a curious binomial transform. In particular, the degrees of these Hankel determinants share the same nature as those in one of the <em>q</em>-binomial cases.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"176 ","pages":"Article 103051"},"PeriodicalIF":1.3,"publicationDate":"2026-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146080818","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 : 2026-01-26DOI: 10.1016/j.aam.2026.103047
J.E. Paguyo
Let X be a finite set and let G be a finite group acting on X. The group action splits X into disjoint orbits. The Burnside process is a Markov chain on X which has a uniform stationary distribution when the chain is lumped to orbits. We consider the case where with and is the symmetric group on , such that G acts on X by permuting the value of each coordinate. The resulting Burnside process gives a novel algorithm for sampling a set partition of uniformly at random. We obtain bounds on the mixing time and show that the chain is rapidly mixing. For the case , the algorithm corresponds to sampling a set partition of with at most k blocks, and we obtain a mixing time bound which is independent of n. 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.
{"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><</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":"2026-01-26","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}
Pub Date : 2026-01-23DOI: 10.1016/j.aam.2026.103049
Mihai Ciucu
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.
{"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":"2026-01-23","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}
Pub Date : 2026-01-23DOI: 10.1016/j.aam.2026.103045
Maya Thompson
The surface Tutte polynomial has recently been generalised to pseudo-surfaces equipping it with recursive deletion-contraction relations [15]. 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 [3], [4], [18], which we recover from our main result.
曲面Tutte多项式最近被推广到具有递归删缩关系[15]的伪曲面。我们使用这些关系来证明这个推广自然具有拟树展开式。这扩展了Bollobás-Riordan， Las Vergnas和Krushkal多项式[3]，[4]，[18]的拟树展开，我们从我们的主要结果中恢复。
{"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":"2026-01-23","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}
Pub Date : 2026-01-23DOI: 10.1016/j.aam.2026.103048
Deke Zhao , Zhankui Xiao
The paper aims to provide a categorification of the Brenti–Welker identity involving Eulerian numbers in (Adv. Appl. Math. 42 (2009): 545–556) by lifting it from an enumerative equality to an isomorphism of symmetric group representations. To do so, we study the decomposition of the tensor product of and modules affording Foulkes characters as modules of the symmetric group. The main ingredient of the proof is a combinatorial identity which may be of independent interest.
{"title":"A categorification of the Brenti–Welker identity","authors":"Deke Zhao , Zhankui Xiao","doi":"10.1016/j.aam.2026.103048","DOIUrl":"10.1016/j.aam.2026.103048","url":null,"abstract":"<div><div>The paper aims to provide a categorification of the Brenti–Welker identity involving Eulerian numbers in (Adv. Appl. Math. 42 (2009): 545–556) by lifting it from an enumerative equality to an isomorphism of symmetric group representations. To do so, we study the decomposition of the tensor product of <span><math><msup><mrow><mo>(</mo><msup><mrow><mi>C</mi></mrow><mrow><mi>r</mi></mrow></msup><mo>)</mo></mrow><mrow><mo>⊗</mo><mi>n</mi></mrow></msup></math></span> and modules affording Foulkes characters as modules of the symmetric group. The main ingredient of the proof is a combinatorial identity which may be of independent interest.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"176 ","pages":"Article 103048"},"PeriodicalIF":1.3,"publicationDate":"2026-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039693","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 : 2026-01-20DOI: 10.1016/j.aam.2025.103021
Pedro Ribeiro
Let denote the number of representations of the positive integer n as the sum of k squares. We prove a new summation formula involving and the Bessel functions of the first kind, which constitutes an analogue of a result due to the Russian mathematician A. I. Popov.
{"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":"2026-01-20","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}
Pub Date : 2026-01-19DOI: 10.1016/j.aam.2026.103043
Eugene Strahov
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.
{"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":"2026-01-19","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}
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