{"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}
{"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}
{"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}
{"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}
{"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}
{"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}
{"title":"k-loose elements and k-paving matroids","authors":"Jagdeep Singh","doi":"10.1016/j.aam.2025.102885","DOIUrl":"10.1016/j.aam.2025.102885","url":null,"abstract":"<div><div>For a matroid of rank <em>r</em> and a non-negative integer <em>k</em>, an element is called <em>k</em>-loose if every circuit containing it has size greater than <span><math><mi>r</mi><mo>−</mo><mi>k</mi></math></span>. Zaslavsky and the author characterized all binary matroids with a 1-loose element. In this paper, we establish a sharp linear bound on the size of a binary matroid, in terms of its rank, that contains a <em>k</em>-loose element. A matroid is called <em>k</em>-paving if all its elements are <em>k</em>-loose. Rajpal showed that for a prime power <em>q</em>, the rank of a <span><math><mi>G</mi><mi>F</mi><mo>(</mo><mi>q</mi><mo>)</mo></math></span>-matroid that is <em>k</em>-paving is bounded. We provide a bound on the rank of <span><math><mi>G</mi><mi>F</mi><mo>(</mo><mi>q</mi><mo>)</mo></math></span>-matroids that are cosimple and have two <em>k</em>-loose elements. Consequently, we strengthen the result of Rajpal by providing a bound on the rank of <span><math><mi>G</mi><mi>F</mi><mo>(</mo><mi>q</mi><mo>)</mo></math></span>-matroids that are <em>k</em>-paving. Additionally, we provide a bound on the size of binary matroids that are <em>k</em>-paving.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"167 ","pages":"Article 102885"},"PeriodicalIF":1.0,"publicationDate":"2025-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143644876","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":"Polynomial resultants and Ramsey numbers of a theta graph","authors":"Meng Liu , Ye Wang","doi":"10.1016/j.aam.2025.102881","DOIUrl":"10.1016/j.aam.2025.102881","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>θ</mi></mrow><mrow><mn>3</mn><mo>,</mo><mn>4</mn></mrow></msub></math></span> be the graph consisting of three internally disjoint paths of length four sharing common endpoints. It is shown <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><msub><mrow><mi>θ</mi></mrow><mrow><mn>3</mn><mo>,</mo><mn>4</mn></mrow></msub><mo>)</mo><mo>=</mo><mi>Θ</mi><mo>(</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>4</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>)</mo></math></span> as <span><math><mi>k</mi><mo>→</mo><mo>∞</mo></math></span> by computing polynomial resultants.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"167 ","pages":"Article 102881"},"PeriodicalIF":1.0,"publicationDate":"2025-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143642896","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":"Linear orbits of smooth quadric surfaces","authors":"Franquiz Caraballo Alba","doi":"10.1016/j.aam.2025.102880","DOIUrl":"10.1016/j.aam.2025.102880","url":null,"abstract":"<div><div>The <em>linear orbit</em> of a degree <em>d</em> hypersurface in <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> is its orbit under the natural action of <span><math><mi>P</mi><mi>GL</mi><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>, in the projective space of dimension <span><math><mi>N</mi><mo>=</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>+</mo><mi>d</mi></mrow></mtd></mtr><mtr><mtd><mi>d</mi></mtd></mtr></mtable><mo>)</mo></mrow><mo>−</mo><mn>1</mn></math></span> parameterizing such hypersurfaces. This action restricted to a specific hypersurface <em>X</em> extends to a rational map from the projectivization of the space of matrices to <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span>. The class of the graph of this map is the <em>predegree polynomial</em> of its corresponding hypersurface. The objective of this paper is threefold. First, we formally define the predegree polynomial of a hypersurface in <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, introduced in the case of plane curves by Aluffi and Faber, and prove some results in the general case. A key result in the general setting is that a partial resolution of said rational map can contain enough information to compute the predegree polynomial of a hypersurface. Second, we compute the leading term of the predegree polynomial of a smooth quadric in <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> over an algebraically closed field with characteristic 0, and compute the other coefficients in the specific case <span><math><mi>n</mi><mo>=</mo><mn>3</mn></math></span>. In analogy to Aluffi and Faber's work, the tool for computing this invariant is producing a (partial) resolution of the previously mentioned rational map which contains enough information to obtain the invariant. Third, we provide a complete resolution of the rational map in the case <span><math><mi>n</mi><mo>=</mo><mn>3</mn></math></span>, which in principle could be used to compute more refined invariants.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"167 ","pages":"Article 102880"},"PeriodicalIF":1.0,"publicationDate":"2025-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143636655","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":"Rotational hypersurfaces family satisfying Ln−3G=AG in the n-dimensional Euclidean space","authors":"Erhan Güler , Nurettin Cenk Turgay","doi":"10.1016/j.aam.2025.102879","DOIUrl":"10.1016/j.aam.2025.102879","url":null,"abstract":"<div><div>In this paper, we investigate rotational hypersurfaces family in <em>n</em>-dimensional Euclidean space <span><math><msup><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. Our focus is on studying the Gauss map <span><math><mi>G</mi></math></span> of this family with respect to the operator <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>, which acts on functions defined on the hypersurfaces. The operator <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> can be viewed as a modified Laplacian and is known by various names, including the Cheng–Yau operator in certain cases. Specifically, we focus on the scenario where <span><math><mi>k</mi><mo>=</mo><mi>n</mi><mo>−</mo><mn>3</mn></math></span> and <span><math><mi>n</mi><mo>≥</mo><mn>3</mn></math></span>. By applying the operator <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>3</mn></mrow></msub></math></span> to the Gauss map <span><math><mi>G</mi></math></span>, we establish a classification theorem. This theorem establishes a connection between the <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrix <span><math><mi>A</mi></math></span>, and the Gauss map <span><math><mi>G</mi></math></span> through the equation <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>3</mn></mrow></msub><mi>G</mi><mo>=</mo><mi>A</mi><mi>G</mi></math></span>.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"167 ","pages":"Article 102879"},"PeriodicalIF":1.0,"publicationDate":"2025-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143636651","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}
京公网安备 11010802042870号
求助内容：
应助结果提醒方式：