Pub Date : 2025-04-14DOI: 10.1016/j.aam.2025.102901
Houshan Fu , Xiangyu Ren , Suijie Wang
<div><div>Kochol introduced the assigning polynomial <span><math><mi>F</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>α</mi><mo>;</mo><mi>k</mi><mo>)</mo></math></span> to count nowhere-zero <span><math><mo>(</mo><mi>A</mi><mo>,</mo><mi>b</mi><mo>)</mo></math></span>-flows of a graph <em>G</em>, where <em>A</em> is a finite Abelian group and <em>α</em> is a <span><math><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span>-assigning from a family <span><math><mi>Λ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of certain nonempty vertex subsets of <em>G</em> to <span><math><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span>. We introduce the concepts of <em>b</em>-compatible graph and <em>b</em>-compatible broken bond to give an explicit formula for the assigning polynomials and to examine their coefficients. More specifically, for a function <span><math><mi>b</mi><mo>:</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>→</mo><mi>A</mi></math></span>, let <span><math><msub><mrow><mi>α</mi></mrow><mrow><mi>G</mi><mo>,</mo><mi>b</mi></mrow></msub></math></span> be a <span><math><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span>-assigning of <em>G</em> such that for each <span><math><mi>X</mi><mo>∈</mo><mi>Λ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, <span><math><msub><mrow><mi>α</mi></mrow><mrow><mi>G</mi><mo>,</mo><mi>b</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span> if and only if <span><math><msub><mrow><mo>∑</mo></mrow><mrow><mi>v</mi><mo>∈</mo><mi>X</mi></mrow></msub><mi>b</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span>. We show that for any <span><math><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span>-assigning <em>α</em> of <em>G</em>, if there exists a function <span><math><mi>b</mi><mo>:</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>→</mo><mi>A</mi></math></span> such that <em>G</em> is <em>b</em>-compatible and <span><math><mi>α</mi><mo>=</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>G</mi><mo>,</mo><mi>b</mi></mrow></msub></math></span>, then the assigning polynomial <span><math><mi>F</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>α</mi><mo>;</mo><mi>k</mi><mo>)</mo></math></span> has the <em>b</em>-compatible spanning subgraph expansion<span><span><span><math><mi>F</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>α</mi><mo>;</mo><mi>k</mi><mo>)</mo><mo>=</mo><munder><mo>∑</mo><mrow><mtable><mtr><mtd><mi>S</mi><mo>⊆</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>,</mo></mtd></mtr><mtr><mtd><mi>G</mi><mo>−</mo><mi>S</mi><mrow><mtext> is</mtext><mspace></mspace><mtext>b</mtext><mtext>-compatible</mtext></mrow></mtd></mtr></mtable></mrow></munder><msup><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>|</mo><mi>S</mi><mo>|</mo></mrow></msup><msup><mrow><mi>k</mi></mrow><mrow><mi>m</mi><mo>(</mo><mi>G</mi><mo>−</mo><mi>S</mi><mo>)</mo></mrow></msup><mo>,</mo></math></span></span></span> and is the following form<span><span><span><math><mi>F</mi>
{"title":"Counting flows of b-compatible graphs","authors":"Houshan Fu , Xiangyu Ren , Suijie Wang","doi":"10.1016/j.aam.2025.102901","DOIUrl":"10.1016/j.aam.2025.102901","url":null,"abstract":"<div><div>Kochol introduced the assigning polynomial <span><math><mi>F</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>α</mi><mo>;</mo><mi>k</mi><mo>)</mo></math></span> to count nowhere-zero <span><math><mo>(</mo><mi>A</mi><mo>,</mo><mi>b</mi><mo>)</mo></math></span>-flows of a graph <em>G</em>, where <em>A</em> is a finite Abelian group and <em>α</em> is a <span><math><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span>-assigning from a family <span><math><mi>Λ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of certain nonempty vertex subsets of <em>G</em> to <span><math><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span>. We introduce the concepts of <em>b</em>-compatible graph and <em>b</em>-compatible broken bond to give an explicit formula for the assigning polynomials and to examine their coefficients. More specifically, for a function <span><math><mi>b</mi><mo>:</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>→</mo><mi>A</mi></math></span>, let <span><math><msub><mrow><mi>α</mi></mrow><mrow><mi>G</mi><mo>,</mo><mi>b</mi></mrow></msub></math></span> be a <span><math><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span>-assigning of <em>G</em> such that for each <span><math><mi>X</mi><mo>∈</mo><mi>Λ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, <span><math><msub><mrow><mi>α</mi></mrow><mrow><mi>G</mi><mo>,</mo><mi>b</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span> if and only if <span><math><msub><mrow><mo>∑</mo></mrow><mrow><mi>v</mi><mo>∈</mo><mi>X</mi></mrow></msub><mi>b</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span>. We show that for any <span><math><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span>-assigning <em>α</em> of <em>G</em>, if there exists a function <span><math><mi>b</mi><mo>:</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>→</mo><mi>A</mi></math></span> such that <em>G</em> is <em>b</em>-compatible and <span><math><mi>α</mi><mo>=</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>G</mi><mo>,</mo><mi>b</mi></mrow></msub></math></span>, then the assigning polynomial <span><math><mi>F</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>α</mi><mo>;</mo><mi>k</mi><mo>)</mo></math></span> has the <em>b</em>-compatible spanning subgraph expansion<span><span><span><math><mi>F</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>α</mi><mo>;</mo><mi>k</mi><mo>)</mo><mo>=</mo><munder><mo>∑</mo><mrow><mtable><mtr><mtd><mi>S</mi><mo>⊆</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>,</mo></mtd></mtr><mtr><mtd><mi>G</mi><mo>−</mo><mi>S</mi><mrow><mtext> is</mtext><mspace></mspace><mtext>b</mtext><mtext>-compatible</mtext></mrow></mtd></mtr></mtable></mrow></munder><msup><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>|</mo><mi>S</mi><mo>|</mo></mrow></msup><msup><mrow><mi>k</mi></mrow><mrow><mi>m</mi><mo>(</mo><mi>G</mi><mo>−</mo><mi>S</mi><mo>)</mo></mrow></msup><mo>,</mo></math></span></span></span> and is the following form<span><span><span><math><mi>F</mi>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"168 ","pages":"Article 102901"},"PeriodicalIF":1.0,"publicationDate":"2025-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143826047","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-04-14DOI: 10.1016/j.aam.2025.102888
Honglin Zhu
Given a grid polygon P in a grid of equilateral triangles, Defant and Jiradilok considered a billiards system where beams of light bounce around inside P. We study the relationship between the perimeter of P and the number of different trajectories that the billiards system has. Resolving a conjecture of Defant and Jiradilok, we prove the sharp inequality and characterize the equality cases.
{"title":"The maximum number of cycles in a triangular-grid billiards system with a given perimeter","authors":"Honglin Zhu","doi":"10.1016/j.aam.2025.102888","DOIUrl":"10.1016/j.aam.2025.102888","url":null,"abstract":"<div><div>Given a grid polygon <em>P</em> in a grid of equilateral triangles, Defant and Jiradilok considered a billiards system where beams of light bounce around inside <em>P</em>. We study the relationship between the perimeter <span><math><mi>perim</mi><mo>(</mo><mi>P</mi><mo>)</mo></math></span> of <em>P</em> and the number of different trajectories <span><math><mi>cyc</mi><mo>(</mo><mi>P</mi><mo>)</mo></math></span> that the billiards system has. Resolving a conjecture of Defant and Jiradilok, we prove the sharp inequality <span><math><mi>cyc</mi><mo>(</mo><mi>P</mi><mo>)</mo><mo>≤</mo><mo>(</mo><mi>perim</mi><mo>(</mo><mi>P</mi><mo>)</mo><mo>+</mo><mn>2</mn><mo>)</mo><mo>/</mo><mn>4</mn></math></span> and characterize the equality cases.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"168 ","pages":"Article 102888"},"PeriodicalIF":1.0,"publicationDate":"2025-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143826046","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 : 2025-04-10DOI: 10.1016/j.aam.2025.102889
Sen-Peng Eu , Louis Kao , Juei-Yin Lin
Cai and Readdy proposed a new framework for studying the q-analogue of a combinatorial structure S. Specifically, the aim is to identify two statistics over S and a proper subset of S such that represents the q--expansion over , and to explore the poset and topological interpretations of this expansion. Cai and Readdy provided comprehensive profiles for classical Stirling numbers of both kinds within this framework. In this work, we extend Cai and Readdy's results to colored q-Stirling numbers of both kinds, as well as colored q-Lah numbers. We also briefly discuss q-Stirling and q-Lah numbers of type D.
蔡和瑞迪提出了一个研究组合结构 S 的 q-analogue f(q) 的新框架。具体来说，其目的是找出 S 上的两个统计量和 S 的一个适当子集 S′，从而使 f(q) 代表 S′上的 q-(1+q)- 展开，并探索这种展开的正集和拓扑解释。Cai 和 Readdy 在此框架内提供了两种经典斯特林数的全面剖面图。在这项工作中，我们将蔡和雷迪的结果扩展到两种彩色 q-Stirling 数以及彩色 q-Lah 数。我们还简要讨论了 D 型的 q-Stirling 和 q-Lah 数。
{"title":"Colored q-Stirling and q-Lah numbers: A new view continued","authors":"Sen-Peng Eu , Louis Kao , Juei-Yin Lin","doi":"10.1016/j.aam.2025.102889","DOIUrl":"10.1016/j.aam.2025.102889","url":null,"abstract":"<div><div>Cai and Readdy proposed a new framework for studying the <em>q</em>-analogue <span><math><mi>f</mi><mo>(</mo><mi>q</mi><mo>)</mo></math></span> of a combinatorial structure <em>S</em>. Specifically, the aim is to identify two statistics over <em>S</em> and a proper subset <span><math><msup><mrow><mi>S</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> of <em>S</em> such that <span><math><mi>f</mi><mo>(</mo><mi>q</mi><mo>)</mo></math></span> represents the <em>q</em>-<span><math><mo>(</mo><mn>1</mn><mo>+</mo><mi>q</mi><mo>)</mo></math></span>-expansion over <span><math><msup><mrow><mi>S</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span>, and to explore the poset and topological interpretations of this expansion. Cai and Readdy provided comprehensive profiles for classical Stirling numbers of both kinds within this framework. In this work, we extend Cai and Readdy's results to colored <em>q</em>-Stirling numbers of both kinds, as well as colored <em>q</em>-Lah numbers. We also briefly discuss <em>q</em>-Stirling and <em>q</em>-Lah numbers of type <em>D</em>.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"168 ","pages":"Article 102889"},"PeriodicalIF":1.0,"publicationDate":"2025-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143808186","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 : 2025-04-09DOI: 10.1016/j.aam.2025.102890
Irem Portakal , Daniel Windisch
This paper is a significant step forward in understanding dependency equilibria within the framework of real algebraic geometry encompassing both pure and mixed equilibria. In alignment with Spohn's original definition of dependency equilibria, we propose two alternative definitions, allowing for an algebro-geometric comprehensive study of all dependency equilibria. We give a sufficient condition for the existence of a pure dependency equilibrium and show that every Nash equilibrium lies on the Spohn variety, the algebraic model for dependency equilibria. For generic games, the set of real points of the Spohn variety is Zariski dense. Furthermore, every Nash equilibrium in this case is a dependency equilibrium. Finally, we present a detailed analysis of the geometric structure of dependency equilibria for -games.
{"title":"Dependency equilibria: Boundary cases and their real algebraic geometry","authors":"Irem Portakal , Daniel Windisch","doi":"10.1016/j.aam.2025.102890","DOIUrl":"10.1016/j.aam.2025.102890","url":null,"abstract":"<div><div>This paper is a significant step forward in understanding dependency equilibria within the framework of real algebraic geometry encompassing both pure and mixed equilibria. In alignment with Spohn's original definition of dependency equilibria, we propose two alternative definitions, allowing for an algebro-geometric comprehensive study of all dependency equilibria. We give a sufficient condition for the existence of a pure dependency equilibrium and show that every Nash equilibrium lies on the Spohn variety, the algebraic model for dependency equilibria. For generic games, the set of real points of the Spohn variety is Zariski dense. Furthermore, every Nash equilibrium in this case is a dependency equilibrium. Finally, we present a detailed analysis of the geometric structure of dependency equilibria for <span><math><mo>(</mo><mn>2</mn><mo>×</mo><mn>2</mn><mo>)</mo></math></span>-games.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"168 ","pages":"Article 102890"},"PeriodicalIF":1.0,"publicationDate":"2025-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143808185","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 : 2025-04-08DOI: 10.1016/j.aam.2025.102899
Colin Defant, Mitchell Lee
Defant found that the relationship between a sequence of (univariate) classical cumulants and the corresponding sequence of (univariate) free cumulants can be described combinatorially in terms of families of binary plane trees called troupes. Using a generalization of troupes that we call weighted troupes, we generalize this result to allow for multivariate cumulants. Our result also gives a combinatorial description of the corresponding Boolean cumulants. This allows us to answer a question of Defant regarding his troupe transform. We also provide explicit distributions whose cumulants correspond to some specific weighted troupes.
{"title":"Boolean, free, and classical cumulants as tree enumerations","authors":"Colin Defant, Mitchell Lee","doi":"10.1016/j.aam.2025.102899","DOIUrl":"10.1016/j.aam.2025.102899","url":null,"abstract":"<div><div>Defant found that the relationship between a sequence of (univariate) classical cumulants and the corresponding sequence of (univariate) free cumulants can be described combinatorially in terms of families of binary plane trees called <em>troupes</em>. Using a generalization of troupes that we call <em>weighted troupes</em>, we generalize this result to allow for multivariate cumulants. Our result also gives a combinatorial description of the corresponding Boolean cumulants. This allows us to answer a question of Defant regarding his <em>troupe transform</em>. We also provide explicit distributions whose cumulants correspond to some specific weighted troupes.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"168 ","pages":"Article 102899"},"PeriodicalIF":1.0,"publicationDate":"2025-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143792352","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 : 2025-04-04DOI: 10.1016/j.aam.2025.102887
Farid Aliniaeifard, Stephanie van Willigenburg
We answer a question of Bergeron, Hohlweg, Rosas, and Zabrocki from 2006 to give a combinatorial description for the coproduct of the x-basis in the Hopf algebra of symmetric functions in noncommuting variables, NCSym, which arises in the theory of Grothendieck bialgebras. We achieve this by applying the theory of Hopf monoids and the Fock functor. We also determine combinatorial expansions of this basis in terms of the monomial and power sum symmetric functions in NCSym, and by taking the commutative image of the x-basis we discover a new multiplicative basis for the algebra of symmetric functions.
我们回答了Bergeron， Hohlweg, Rosas， and Zabrocki从2006年提出的一个问题，给出了非交换变量NCSym对称函数的Hopf代数中x基的余积的组合描述，该问题出现在Grothendieck双代数理论中。我们利用Hopf半群理论和Fock函子实现了这一点。我们还用NCSym中的单项式和幂和对称函数确定了该基的组合展开式，并通过取x基的交换像发现了对称函数代数的一种新的乘法基。
{"title":"The extra basis in noncommuting variables","authors":"Farid Aliniaeifard, Stephanie van Willigenburg","doi":"10.1016/j.aam.2025.102887","DOIUrl":"10.1016/j.aam.2025.102887","url":null,"abstract":"<div><div>We answer a question of Bergeron, Hohlweg, Rosas, and Zabrocki from 2006 to give a combinatorial description for the coproduct of the <em>x</em>-basis in the Hopf algebra of symmetric functions in noncommuting variables, NCSym, which arises in the theory of Grothendieck bialgebras. We achieve this by applying the theory of Hopf monoids and the Fock functor. We also determine combinatorial expansions of this basis in terms of the monomial and power sum symmetric functions in NCSym, and by taking the commutative image of the <em>x</em>-basis we discover a new multiplicative basis for the algebra of symmetric functions.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"168 ","pages":"Article 102887"},"PeriodicalIF":1.0,"publicationDate":"2025-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143767926","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-04-01DOI: 10.1016/j.aam.2025.102886
David G.L. Wang , James Z.F. Zhou
We develop a composition method to unearth positive -expansions of chromatic symmetric functions , where the subscript I stands for compositions rather than integer partitions. Using this method, we derive positive and neat -expansions for the chromatic symmetric functions of tadpoles, barbells and generalized bulls, and establish the e-positivity of hats. We also obtain a compact ribbon Schur analog for the chromatic symmetric function of cycles.
{"title":"A composition method for neat formulas of chromatic symmetric functions","authors":"David G.L. Wang , James Z.F. Zhou","doi":"10.1016/j.aam.2025.102886","DOIUrl":"10.1016/j.aam.2025.102886","url":null,"abstract":"<div><div>We develop a composition method to unearth positive <span><math><msub><mrow><mi>e</mi></mrow><mrow><mi>I</mi></mrow></msub></math></span>-expansions of chromatic symmetric functions <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>G</mi></mrow></msub></math></span>, where the subscript <em>I</em> stands for compositions rather than integer partitions. Using this method, we derive positive and neat <span><math><msub><mrow><mi>e</mi></mrow><mrow><mi>I</mi></mrow></msub></math></span>-expansions for the chromatic symmetric functions of tadpoles, barbells and generalized bulls, and establish the <em>e</em>-positivity of hats. We also obtain a compact ribbon Schur analog for the chromatic symmetric function of cycles.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"167 ","pages":"Article 102886"},"PeriodicalIF":1.0,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143739432","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 : 2025-03-28DOI: 10.1016/j.aam.2025.102883
Persi Diaconis , Nathan Tung
Let be a group of permutations of kn objects which permutes things independently in disjoint blocks of size k and then permutes the blocks. We investigate the probabilistic and enumerative aspects of random elements of . This includes novel limit theorems for cycles of various lengths, number of cycles, and inversions. The limits include compound Poisson distributions with interesting dependence structure.
{"title":"Poisson approximation for large permutation groups","authors":"Persi Diaconis , Nathan Tung","doi":"10.1016/j.aam.2025.102883","DOIUrl":"10.1016/j.aam.2025.102883","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span> be a group of permutations of <em>kn</em> objects which permutes things independently in disjoint blocks of size <em>k</em> and then permutes the blocks. We investigate the probabilistic and enumerative aspects of random elements of <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span>. This includes novel limit theorems for cycles of various lengths, number of cycles, and inversions. The limits include compound Poisson distributions with interesting dependence structure.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"167 ","pages":"Article 102883"},"PeriodicalIF":1.0,"publicationDate":"2025-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143714822","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 : 2025-03-26DOI: 10.1016/j.aam.2025.102884
Xiangyu Ding, Lisa Hui Sun
In 2012, Andrews and Merca obtained a truncated version of Euler's pentagonal number theorem and showed the nonnegativity related to partition functions. Meanwhile, Andrews and Merca, Guo and Zeng independently conjectured that the truncated Jacobi triple product series has nonnegative coefficients, which has been confirmed analytically and also combinatorially. In 2022, Merca proposed a stronger version for this conjecture. In this paper, by applying Agarwal, Andrews and Bressoud's identity derived from the Bailey lattice, we obtain a truncated version for the Jacobi triple product series with odd basis, which reduces to the Andrews–Gordon identity as a special instance. As consequences, we obtain new truncated forms for Euler's pentagonal number theorem, Gauss' theta series on triangular numbers and square numbers, which lead to inequalities for certain partition functions. Moreover, by considering a truncated theta series involving ℓ-regular partitions, we confirm a conjecture proposed by Ballantine and Merca about 6-regular partitions and show that Merca's stronger conjecture on truncated Jacobi triple product series holds when for .
{"title":"Truncated theta series from the Bailey lattice","authors":"Xiangyu Ding, Lisa Hui Sun","doi":"10.1016/j.aam.2025.102884","DOIUrl":"10.1016/j.aam.2025.102884","url":null,"abstract":"<div><div>In 2012, Andrews and Merca obtained a truncated version of Euler's pentagonal number theorem and showed the nonnegativity related to partition functions. Meanwhile, Andrews and Merca, Guo and Zeng independently conjectured that the truncated Jacobi triple product series has nonnegative coefficients, which has been confirmed analytically and also combinatorially. In 2022, Merca proposed a stronger version for this conjecture. In this paper, by applying Agarwal, Andrews and Bressoud's identity derived from the Bailey lattice, we obtain a truncated version for the Jacobi triple product series with odd basis, which reduces to the Andrews–Gordon identity as a special instance. As consequences, we obtain new truncated forms for Euler's pentagonal number theorem, Gauss' theta series on triangular numbers and square numbers, which lead to inequalities for certain partition functions. Moreover, by considering a truncated theta series involving <em>ℓ</em>-regular partitions, we confirm a conjecture proposed by Ballantine and Merca about 6-regular partitions and show that Merca's stronger conjecture on truncated Jacobi triple product series holds when <span><math><mi>R</mi><mo>=</mo><mn>3</mn><mi>S</mi></math></span> for <span><math><mi>S</mi><mo>≥</mo><mn>1</mn></math></span>.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"167 ","pages":"Article 102884"},"PeriodicalIF":1.0,"publicationDate":"2025-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143704752","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 : 2025-03-20DOI: 10.1016/j.aam.2025.102882
Kaimei Huang, Sherry H.F. Yan
<div><div>As natural generalizations of the descent number (<span><math><mi>des</mi></math></span>) and the major index (<span><math><mi>maj</mi></math></span>), Rawlings introduced the notions of the <em>r</em>-descent number (<span><math><mi>r</mi><mrow><mi>des</mi></mrow></math></span>) and the <em>r</em>-major index (<span><math><mi>r</mi><mrow><mi>maj</mi></mrow></math></span>) for a given positive integer <em>r</em>. A pair <span><math><mo>(</mo><msub><mrow><mi>st</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>st</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> of permutation statistics is said to be <em>r</em>-Euler-Mahonian if <span><math><mo>(</mo><mrow><mi>s</mi><msub><mrow><mi>t</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mo>,</mo><mrow><mi>s</mi><msub><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mo>)</mo></math></span> and <span><math><mo>(</mo><mi>r</mi><mrow><mi>des</mi></mrow><mo>,</mo><mi>r</mi><mrow><mi>maj</mi></mrow><mo>)</mo></math></span> are equidistributed over the set <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> of all permutations 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>. The main objective of this paper is to confirm a recent conjecture posed by Liu which asserts that <span><math><mo>(</mo><mi>g</mi><msub><mrow><mi>exc</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>,</mo><mi>g</mi><msub><mrow><mi>den</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></math></span> is <span><math><mo>(</mo><mi>g</mi><mo>+</mo><mi>ℓ</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-Euler-Mahonian for all positive integers <em>g</em> and <em>ℓ</em>, where <span><math><mi>g</mi><msub><mrow><mi>exc</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math></span> denotes the <em>g</em>-gap <em>ℓ</em>-level excedance number and <span><math><mi>g</mi><msub><mrow><mi>den</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math></span> denotes the <em>g</em>-gap <em>ℓ</em>-level Denert's statistic. This is accomplished via a bijective proof of the equidistribution of <span><math><mo>(</mo><mi>g</mi><msub><mrow><mi>exc</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>,</mo><mi>g</mi><msub><mrow><mi>den</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></math></span> and <span><math><mo>(</mo><mi>r</mi><mrow><mi>des</mi></mrow><mo>,</mo><mi>r</mi><mrow><mi>maj</mi></mrow><mo>)</mo></math></span> where <span><math><mi>r</mi><mo>=</mo><mi>g</mi><mo>+</mo><mi>ℓ</mi><mo>−</mo><mn>1</mn></math></span>. Setting <span><math><mi>g</mi><mo>=</mo><mi>ℓ</mi><mo>=</mo><mn>1</mn></math></span>, our result recovers the equidistribution of <span><math><mo>(</mo><mrow><mi>des</mi></mrow><mo>,</mo><mrow><mi>maj</mi></mrow><mo>)</mo></math></span> and <span><math><mo>(</mo><mrow><mi>exc</mi></mrow><mo>,</mo><mrow><mi>den</mi></mrow><mo>)</mo></math></span>, which was first conjectured by Denert and proved by Foata
作为下降数（des）和主索引（maj）的自然推广，Rawlings引入了给定正整数r的r-下降数（rdes）和r-主索引（rmaj）的概念。如果（st1,st2）和（rdes,rmaj）在{1,2，…，n}的所有排列的集合Sn上是均匀分布的，那么一对（st1,st2）排列统计量就是r- euler - mahonian。本文的主要目的是证实Liu最近提出的一个猜想，即对于所有正整数g和r， （gexc r,gden r）是(g+ r−1)-Euler-Mahonian，其中gexc r表示g-gap r -level的超越数，gden r表示g-gap r -level的Denert's统计量。这是通过客观证明（gexc r,gden r）和（rdes,rmaj）的均匀分布来实现的，其中r=g+ r−1。设g= r =1，我们的结果恢复了（des,maj）和（exc,den）的均匀分布，这是由Denert首先推测并由Foata和Zeilberger证明的。我们的第二个主要结果与（geexc r,gdeng+ r）的类似结果有关，该结果表明（geexc r,gdeng+ r）对于所有正整数g和r都是(g+ r−1)-欧拉-马霍尼量。
{"title":"Further results on r-Euler-Mahonian statistics","authors":"Kaimei Huang, Sherry H.F. Yan","doi":"10.1016/j.aam.2025.102882","DOIUrl":"10.1016/j.aam.2025.102882","url":null,"abstract":"<div><div>As natural generalizations of the descent number (<span><math><mi>des</mi></math></span>) and the major index (<span><math><mi>maj</mi></math></span>), Rawlings introduced the notions of the <em>r</em>-descent number (<span><math><mi>r</mi><mrow><mi>des</mi></mrow></math></span>) and the <em>r</em>-major index (<span><math><mi>r</mi><mrow><mi>maj</mi></mrow></math></span>) for a given positive integer <em>r</em>. A pair <span><math><mo>(</mo><msub><mrow><mi>st</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>st</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> of permutation statistics is said to be <em>r</em>-Euler-Mahonian if <span><math><mo>(</mo><mrow><mi>s</mi><msub><mrow><mi>t</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mo>,</mo><mrow><mi>s</mi><msub><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mo>)</mo></math></span> and <span><math><mo>(</mo><mi>r</mi><mrow><mi>des</mi></mrow><mo>,</mo><mi>r</mi><mrow><mi>maj</mi></mrow><mo>)</mo></math></span> are equidistributed over the set <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> of all permutations 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>. The main objective of this paper is to confirm a recent conjecture posed by Liu which asserts that <span><math><mo>(</mo><mi>g</mi><msub><mrow><mi>exc</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>,</mo><mi>g</mi><msub><mrow><mi>den</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></math></span> is <span><math><mo>(</mo><mi>g</mi><mo>+</mo><mi>ℓ</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-Euler-Mahonian for all positive integers <em>g</em> and <em>ℓ</em>, where <span><math><mi>g</mi><msub><mrow><mi>exc</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math></span> denotes the <em>g</em>-gap <em>ℓ</em>-level excedance number and <span><math><mi>g</mi><msub><mrow><mi>den</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math></span> denotes the <em>g</em>-gap <em>ℓ</em>-level Denert's statistic. This is accomplished via a bijective proof of the equidistribution of <span><math><mo>(</mo><mi>g</mi><msub><mrow><mi>exc</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>,</mo><mi>g</mi><msub><mrow><mi>den</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></math></span> and <span><math><mo>(</mo><mi>r</mi><mrow><mi>des</mi></mrow><mo>,</mo><mi>r</mi><mrow><mi>maj</mi></mrow><mo>)</mo></math></span> where <span><math><mi>r</mi><mo>=</mo><mi>g</mi><mo>+</mo><mi>ℓ</mi><mo>−</mo><mn>1</mn></math></span>. Setting <span><math><mi>g</mi><mo>=</mo><mi>ℓ</mi><mo>=</mo><mn>1</mn></math></span>, our result recovers the equidistribution of <span><math><mo>(</mo><mrow><mi>des</mi></mrow><mo>,</mo><mrow><mi>maj</mi></mrow><mo>)</mo></math></span> and <span><math><mo>(</mo><mrow><mi>exc</mi></mrow><mo>,</mo><mrow><mi>den</mi></mrow><mo>)</mo></math></span>, which was first conjectured by Denert and proved by Foata ","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"167 ","pages":"Article 102882"},"PeriodicalIF":1.0,"publicationDate":"2025-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143686514","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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