{"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}
{"title":"The fundamental group in discrete homotopy theory","authors":"Krzysztof Kapulkin, Udit Mavinkurve","doi":"10.1016/j.aam.2024.102838","DOIUrl":"10.1016/j.aam.2024.102838","url":null,"abstract":"<div><div>We develop a robust foundation for studying the fundamental group(oid) in A-homotopy theory, including: equivalent definitions and basic properties, the theory of covering graphs, and the discrete version of the Seifert–van Kampen theorem.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"164 ","pages":"Article 102838"},"PeriodicalIF":1.0,"publicationDate":"2025-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143154466","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}
Shi-Hao Li , Bo-Jian Shen , Guo-Fu Yu , Peter J. Forrester
{"title":"Discrete orthogonal ensemble on the exponential lattices","authors":"Shi-Hao Li , Bo-Jian Shen , Guo-Fu Yu , Peter J. Forrester","doi":"10.1016/j.aam.2024.102836","DOIUrl":"10.1016/j.aam.2024.102836","url":null,"abstract":"<div><div>Inspired by Aomoto's <em>q</em>-Selberg integral, a study is made of an orthogonal ensemble on an exponential lattice. By introducing a skew symmetric kernel, the configuration space of this ensemble is constructed to be symmetric and thus the corresponding skew inner product, skew orthogonal polynomials as well as correlation functions are explicitly formulated. These involve polynomials from the Askey scheme. Examples considered include the Al-Salam & Carlitz, <em>q</em>-Laguerre, little <em>q</em>-Jacobi and big <em>q</em>-Jacobi cases.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"164 ","pages":"Article 102836"},"PeriodicalIF":1.0,"publicationDate":"2025-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143154467","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":"Cells of fixed height in Catalan words and restricted growth functions","authors":"Aubrey Blecher, Arnold Knopfmacher","doi":"10.1016/j.aam.2024.102835","DOIUrl":"10.1016/j.aam.2024.102835","url":null,"abstract":"<div><div>A word <span><math><mi>w</mi><mo>=</mo><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>w</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⋯</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> of length <em>n</em> over the set of positive integers is called a Catalan word whenever <span><math><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>1</mn></math></span> and <span><math><mn>1</mn><mo>≤</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><mn>1</mn></math></span> for <span><math><mi>k</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi></math></span>. A restricted growth function is defined as a word <span><math><mi>w</mi><mo>=</mo><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>w</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⋯</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> of length <em>n</em> over the set of positive integers where <span><math><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>1</mn></math></span> and for <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> we have <span><math><mn>1</mn><mo>≤</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>≤</mo><mi>max</mi><mo></mo><mo>{</mo><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>w</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>}</mo><mo>+</mo><mn>1</mn></math></span>. We also define cells and heights of cells and we represent such words as bargraphs (otherwise known as polyominoes) where the <em>i</em>th column contains <span><math><msub><mrow><mi>w</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> cells for <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi></math></span> and where all columns have their bottom cell on the <em>x</em>-axis. In the case of Catalan words, we prove a relationship between the number of cells at different heights and first terms of the expanded polynomial <span><math><msup><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo>)</mo></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msup></math></span>. In the case of restricted growth functions we find polynomials <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span> where the coefficient of <span><math><msup><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msup></math></span> counts the number of cells of height <em>j</em> across all rgfs with <em>n</em> parts. In this case we also find bivariate generating functions for rgfs with <em>k</em> blocks, where the generating functions tracks the number of cells at a given height as well as the number of parts.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"164 ","pages":"Article 102835"},"PeriodicalIF":1.0,"publicationDate":"2024-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143154472","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":"Conditions for virtually Cohen–Macaulay simplicial complexes","authors":"Adam Van Tuyl , Jay Yang","doi":"10.1016/j.aam.2024.102830","DOIUrl":"10.1016/j.aam.2024.102830","url":null,"abstract":"<div><div>A simplicial complex Δ is a virtually Cohen–Macaulay simplicial complex if its associated Stanley-Reisner ring <em>S</em> has a virtual resolution, as defined by Berkesch, Erman, and Smith, of length <span><math><mrow><mi>codim</mi></mrow><mo>(</mo><mi>S</mi><mo>)</mo></math></span>. We provide a sufficient condition on Δ to be a virtually Cohen–Macaulay simplicial complex. We also introduce virtually shellable simplicial complexes, a generalization of shellable simplicial complexes. Virtually shellable complexes have the property that they are virtually Cohen–Macaulay, generalizing the well-known fact that shellable simplicial complexes are Cohen–Macaulay.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"164 ","pages":"Article 102830"},"PeriodicalIF":1.0,"publicationDate":"2024-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143154474","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":"Dynamics of pop-tsack torsing","authors":"Anqi Li","doi":"10.1016/j.aam.2024.102826","DOIUrl":"10.1016/j.aam.2024.102826","url":null,"abstract":"<div><div>For a finite irreducible Coxeter group <span><math><mo>(</mo><mi>W</mi><mo>,</mo><mi>S</mi><mo>)</mo></math></span> with a fixed Coxeter element <em>c</em> and set of reflections <em>T</em>, Defant and Williams define a pop-tsack torsing operation <span><math><mrow><mi>Po</mi><msub><mrow><mi>p</mi></mrow><mrow><mi>T</mi></mrow></msub></mrow><mo>:</mo><mi>W</mi><mo>→</mo><mi>W</mi></math></span> given by <span><math><mrow><mi>Po</mi><msub><mrow><mi>p</mi></mrow><mrow><mi>T</mi></mrow></msub></mrow><mo>(</mo><mi>w</mi><mo>)</mo><mo>=</mo><mi>w</mi><mo>⋅</mo><mi>π</mi><msup><mrow><mo>(</mo><mi>w</mi><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span> where <span><math><mi>π</mi><mo>(</mo><mi>w</mi><mo>)</mo><mo>=</mo><msubsup><mrow><mo>⋁</mo></mrow><mrow><mi>t</mi><msub><mrow><mo>≤</mo></mrow><mrow><mi>T</mi></mrow></msub><mi>w</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>∈</mo><mi>T</mi></mrow><mrow><mi>N</mi><mi>C</mi><mo>(</mo><mi>w</mi><mo>,</mo><mi>c</mi><mo>)</mo></mrow></msubsup><mi>t</mi></math></span> is the join of all reflections lying below <em>w</em> in the absolute order in the non-crossing partition lattice <span><math><mi>N</mi><mi>C</mi><mo>(</mo><mi>w</mi><mo>,</mo><mi>c</mi><mo>)</mo></math></span>. This is a “dual” notion of the pop-stack sorting operator <span><math><mrow><mi>Po</mi><msub><mrow><mi>p</mi></mrow><mrow><mi>S</mi></mrow></msub></mrow></math></span>; <span><math><mrow><mi>Po</mi><msub><mrow><mi>p</mi></mrow><mrow><mi>S</mi></mrow></msub></mrow></math></span> was introduced by Defant as a way to generalize the pop-stack sorting operator on <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> to general Coxeter groups. Define the forward orbit of an element <span><math><mi>w</mi><mo>∈</mo><mi>W</mi></math></span> to be <span><math><msub><mrow><mi>O</mi></mrow><mrow><mrow><mi>Po</mi><msub><mrow><mi>p</mi></mrow><mrow><mi>T</mi></mrow></msub></mrow></mrow></msub><mo>(</mo><mi>w</mi><mo>)</mo><mo>=</mo><mo>{</mo><mi>w</mi><mo>,</mo><mrow><mi>Po</mi><msub><mrow><mi>p</mi></mrow><mrow><mi>T</mi></mrow></msub></mrow><mo>(</mo><mi>w</mi><mo>)</mo><mo>,</mo><msup><mrow><mi>Po</mi><msub><mrow><mi>p</mi></mrow><mrow><mi>T</mi></mrow></msub></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>w</mi><mo>)</mo><mo>,</mo><mo>…</mo><mo>}</mo></math></span>. Defant and Williams established the length of the longest possible forward orbits <span><math><msub><mrow><mi>max</mi></mrow><mrow><mi>w</mi><mo>∈</mo><mi>W</mi></mrow></msub><mo></mo><mo>|</mo><msub><mrow><mi>O</mi></mrow><mrow><mrow><mi>Po</mi><msub><mrow><mi>p</mi></mrow><mrow><mi>T</mi></mrow></msub></mrow></mrow></msub><mo>(</mo><mi>w</mi><mo>)</mo><mo>|</mo></math></span> for Coxeter groups of coincidental types and Type D in terms of the corresponding Coxeter number of the group. In their paper, they also proposed multiple conjectures about enumerating elements with near maximal orbit length. We resolve all the ","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"164 ","pages":"Article 102826"},"PeriodicalIF":1.0,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143154477","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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