Janet J.W. Dong , Lora R. Du , Kathy Q. Ji , Dax T.X. Zhang
{"title":"New refinements of Narayana polynomials and Motzkin polynomials","authors":"Janet J.W. Dong , Lora R. Du , Kathy Q. Ji , Dax T.X. Zhang","doi":"10.1016/j.aam.2025.102855","DOIUrl":"10.1016/j.aam.2025.102855","url":null,"abstract":"<div><div>Chen, Deutsch and Elizalde introduced a refinement of the Narayana polynomials by distinguishing between old (leftmost child) and young leaves of plane trees. They also provided a refinement of Coker's formula by constructing a bijection. In fact, Coker's formula establishes a connection between the Narayana polynomials and the Motzkin polynomials, which implies the <em>γ</em>-positivity of the Narayana polynomials. In this paper, we introduce the polynomial <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>11</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>12</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>;</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>11</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>12</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>, which further refines the Narayana polynomials by considering leaves of plane trees that have no siblings. We obtain the generating function for <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>11</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>12</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>;</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>11</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>12</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>. To achieve further refinement of Coker's formula based on the polynomial <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>11</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>12</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>;</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>11</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>12</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>, we consider a refinement <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>;</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> of the Motzkin polynomials by classifying the old leaves of a tip-augmented plane tree into three categories and the young leaves into two categories. The generating function for <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msub><mrow><mi>u</mi></mrow><mrow><mn","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"166 ","pages":"Article 102855"},"PeriodicalIF":1.0,"publicationDate":"2025-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143387108","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":"Computing triple and double Hurwitz numbers involving a branch point with a two-part profile","authors":"Zi-Wei Bai, Ricky X.F. Chen","doi":"10.1016/j.aam.2025.102854","DOIUrl":"10.1016/j.aam.2025.102854","url":null,"abstract":"<div><div>The study of Hurwitz numbers intersects with many research areas including representation theory, algebraic geometry and mathematical physics. Though many beautiful general properties have been discovered, obtaining explicit elementary expressions computing these numbers is hard and pertains to a primary goal of the topic. In fact, known explicit formulas are mainly for Hurwitz numbers involving at most two nonsimple branch points (i.e., double Hurwitz numbers). Even for double Hurwitz numbers, only the case where one of the branch points is fully ramified (i.e., one-part double Hurwitz numbers) has been completely and explicitly determined. In this paper, we contribute explicit elementary formulas computing Hurwitz numbers with completed <em>r</em>-cycles involving up to three nonsimple branch points where one of them has a two-part profile, enriching several lines of researches. In particular, we discuss the piecewise polynomiality and the genus-zero case in detail.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"165 ","pages":"Article 102854"},"PeriodicalIF":1.0,"publicationDate":"2025-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143155600","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":"Generating functions of multiple t-star values of general level","authors":"Zhonghua Li, Lu Yan","doi":"10.1016/j.aam.2025.102853","DOIUrl":"10.1016/j.aam.2025.102853","url":null,"abstract":"<div><div>In this paper, we study the explicit expressions of multiple <em>t</em>-star values with an arbitrary number of blocks of twos of general level. We give an expression of a generating function of such values, which generalizes the results for multiple zeta-star values and multiple <em>t</em>-star values. This derived generating function can provide expressions of multiple <em>t</em>-star values of general level in terms of the alternating multiple <em>t</em>-half values of general level with additional factorial and pochhammer symbol. As applications, some specific evaluations of multiple <em>t</em>-star values of general level with one-two-three or more general indices are given. These evaluations contribute to a deeper understanding of the properties of multiple <em>t</em>-star values of general level.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"165 ","pages":"Article 102853"},"PeriodicalIF":1.0,"publicationDate":"2025-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143155599","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}
Ariel Goodwin , Adrian S. Lewis , Genaro López-Acedo , Adriana Nicolae
{"title":"Convex optimization on CAT(0) cubical complexes","authors":"Ariel Goodwin , Adrian S. Lewis , Genaro López-Acedo , Adriana Nicolae","doi":"10.1016/j.aam.2025.102849","DOIUrl":"10.1016/j.aam.2025.102849","url":null,"abstract":"<div><div>We consider geodesically convex optimization problems involving distances to a finite set of points <em>A</em> in a CAT(0) cubical complex. Examples include the minimum enclosing ball problem, the weighted mean and median problems, and the feasibility and projection problems for intersecting balls with centers in <em>A</em>. We propose a decomposition approach relying on standard Euclidean cutting plane algorithms. The cutting planes are readily derivable from efficient algorithms for computing geodesics in the complex.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"165 ","pages":"Article 102849"},"PeriodicalIF":1.0,"publicationDate":"2025-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143155601","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":"Lattice paths and the Rogers–Ramanujan–Gordon type theorems with parity considerations","authors":"Robert X.J. Hao , Diane Y.H. Shi","doi":"10.1016/j.aam.2025.102850","DOIUrl":"10.1016/j.aam.2025.102850","url":null,"abstract":"<div><div>Andrews imposed parity restrictions on the Rogers–Ramanujan–Gordon type partitions, yielding fruitful results. These results were later, advanced by Kurşungöz, Kim, and Yee. In this paper, we construct a bijection between the lattice paths with three types of unitary steps and the Rogers–Ramanujan–Gordon type partitions, which can also provide some refinements of the theorem. By the bijection, we shall give some results involving parity considerations on lattice paths, as the counterpart of Andrews' partition results. Finally, by adding some new restrictions on lattice paths, we also obtain new functions as the generating functions for certain types of lattice paths.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"165 ","pages":"Article 102850"},"PeriodicalIF":1.0,"publicationDate":"2025-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143156472","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":"Combinatorial identities arising from permanents for Euler numbers and Stirling numbers","authors":"Zhicong Lin , Weigen Yan , Tongyuan Zhao","doi":"10.1016/j.aam.2025.102852","DOIUrl":"10.1016/j.aam.2025.102852","url":null,"abstract":"<div><div>We prove several combinatorial identities involving (binomial) Euler numbers and Stirling numbers (in type <em>A</em> or <em>B</em>) of the second kind. These identities arise from our evaluation of the permanents of some special matrices. In particular, a Frobenius-like formula for the 2-Eulerian polynomials is obtained and an alternative approach to Conjecture 1.6 in Fu et al. (2025) <span><span>[8]</span></span> concerning the evaluation of the permanent of the matrix <span><math><msub><mrow><mo>[</mo><mrow><mi>sgn</mi></mrow><mrow><mo>(</mo><mi>cos</mi><mo></mo><mi>π</mi><mfrac><mrow><mi>i</mi><mo>+</mo><mi>j</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>)</mo></mrow><mo>]</mo></mrow><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>≤</mo><mi>n</mi></mrow></msub></math></span> is provided.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"165 ","pages":"Article 102852"},"PeriodicalIF":1.0,"publicationDate":"2025-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143156471","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":"Structural formulas for a family of matrix valued Laguerre polynomials and applications","authors":"Andrea L. Gallo","doi":"10.1016/j.aam.2025.102851","DOIUrl":"10.1016/j.aam.2025.102851","url":null,"abstract":"<div><div>In this work, we study matrix valued orthogonal polynomials (MVOPs) with respect to a Laguerre-type matrix weight. We derive difference-differential relations for these MVOPs and provide explicit expressions for their entries using classical Laguerre polynomials. Under some shifting hypothesis, we demonstrate that the entries of the associated MVOPs can be expressed in terms of dual-Hahn polynomials. Additionally, we give an LDU decomposition for the squared norms of the MVOPs. As an application we study deformed weights and Toda-type equations.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"165 ","pages":"Article 102851"},"PeriodicalIF":1.0,"publicationDate":"2025-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143156469","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":"Twisted cohomology and likelihood ideals","authors":"Saiei-Jaeyeong Matsubara-Heo , Simon Telen","doi":"10.1016/j.aam.2024.102832","DOIUrl":"10.1016/j.aam.2024.102832","url":null,"abstract":"<div><div>A likelihood function on a smooth very affine variety gives rise to a twisted de Rham complex. We show how its top cohomology vector space degenerates to the coordinate ring of the critical points defined by the likelihood equations. We obtain a basis for cohomology from a basis of this coordinate ring. We investigate the dual picture, where twisted cycles correspond to critical points. We show how to expand a twisted cocycle in terms of a basis, and apply our methods to Feynman integrals from physics.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"165 ","pages":"Article 102832"},"PeriodicalIF":1.0,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143156470","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 Orlicz chord Minkowski problem for general measures","authors":"Suwei Li, Qiuyue Chen, Hailin Jin","doi":"10.1016/j.aam.2025.102839","DOIUrl":"10.1016/j.aam.2025.102839","url":null,"abstract":"<div><div>Chord measures and <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> chord measures were recently introduced by Lutwak-Xi-Yang-Zhang by establishing a variational formula regarding a family of fundamental integral geometric invariants called chord integrals. Prescribing the <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> chord measures is called the <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> chord Minkowski problem. The <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> (<span><math><mi>p</mi><mo>></mo><mn>1</mn></math></span>) chord Minkowski problem was solved by Xi-Yang-Zhang-Zhao.</div><div>In the present paper, we investigate the Orlicz chord Minkowski problem, which generalizes the <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub><mspace></mspace><mo>(</mo><mi>p</mi><mo>></mo><mn>1</mn><mo>)</mo></math></span> chord Minkowski problem by replacing <em>p</em> with a fixed decreasing continuous function <span><math><mi>φ</mi><mo>:</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo></mrow><mo>→</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo></mrow></math></span> satisfying <span><math><mi>φ</mi><mo>(</mo><msup><mrow><mn>0</mn></mrow><mrow><mo>+</mo></mrow></msup><mo>)</mo><mo>=</mo><mo>∞</mo></math></span> and <span><math><mi>φ</mi><mo>(</mo><mo>∞</mo><mo>)</mo><mo>=</mo><mn>0</mn></math></span>, and solve the Orlicz chord Minkowski problem for discrete measures and the general measures.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"165 ","pages":"Article 102839"},"PeriodicalIF":1.0,"publicationDate":"2025-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143155594","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":"Random walks, equidistribution and graphical designs","authors":"Stefan Steinerberger, Rekha R. Thomas","doi":"10.1016/j.aam.2024.102837","DOIUrl":"10.1016/j.aam.2024.102837","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></math></span> be a <em>d</em>-regular graph on <em>n</em> vertices and let <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> be a probability measure on <em>V</em>. The act of moving to a randomly chosen neighbor leads to a sequence of probability measures supported on <em>V</em> given by <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mi>A</mi><msup><mrow><mi>D</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><msub><mrow><mi>μ</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>, where <em>A</em> is the adjacency matrix and <em>D</em> is the diagonal matrix of vertex degrees of <em>G</em>. Ordering the eigenvalues of <span><math><mi>A</mi><msup><mrow><mi>D</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span> as <span><math><mn>1</mn><mo>=</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≥</mo><mo>|</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>|</mo><mo>≥</mo><mo>…</mo><mo>≥</mo><mo>|</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>|</mo><mo>≥</mo><mn>0</mn></math></span>, it is well-known that the graphs for which <span><math><mo>|</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>|</mo></math></span> is small are those in which the random walk process converges quickly to the uniform distribution: for all initial probability measures <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> and all <span><math><mi>k</mi><mo>≥</mo><mn>0</mn></math></span>,<span><span><span><math><munder><mo>∑</mo><mrow><mi>v</mi><mo>∈</mo><mi>V</mi></mrow></munder><msup><mrow><mo>|</mo><msub><mrow><mi>μ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>v</mi><mo>)</mo><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>≤</mo><msubsup><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>2</mn><mi>k</mi></mrow></msubsup><mo>.</mo></math></span></span></span> One could wonder whether this rate can be improved for specific initial probability measures <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>. We show that if <em>G</em> is regular, then for any <span><math><mn>1</mn><mo>≤</mo><mi>ℓ</mi><mo>≤</mo><mi>n</mi></math></span>, there exists a probability measure <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> supported on at most <em>ℓ</em> vertices so that<span><span><span><math><munder><mo>∑</mo><mrow><mi>v</mi><mo>∈</mo><mi>V</mi></mrow></munder><msup><mrow><mo>|</mo><msub><mrow><mi>μ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>v</mi><mo>)</mo><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>≤</mo><msubsup><mrow><mi>λ</mi></mro","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"165 ","pages":"Article 102837"},"PeriodicalIF":1.0,"publicationDate":"2025-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143156468","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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