{"title":"d-Separated permutations and q-Stirling numbers of the first kind","authors":"Rosena R.X. Du, Yun Li","doi":"10.1016/j.aam.2025.102976","DOIUrl":"10.1016/j.aam.2025.102976","url":null,"abstract":"<div><div>Let <em>d</em> be a nonnegative integer, a <em>d</em>-separated permutation is a permutation in which every two left-to-right minima are at distance greater than <em>d</em>. More precisely, for <span><math><mi>π</mi><mo>=</mo><msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⋯</mo><msub><mrow><mi>π</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>∈</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, suppose that <span><math><msub><mrow><mi>π</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>,</mo><msub><mrow><mi>π</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>π</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msub></math></span> are the left-to-right minima of <em>π</em> with <span><math><mi>k</mi><mo>≥</mo><mn>1</mn></math></span> and <span><math><mn>1</mn><mo>=</mo><msub><mrow><mi>i</mi></mrow><mrow><mn>1</mn></mrow></msub><mo><</mo><msub><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msub><mo><</mo><mo>⋯</mo><mo><</mo><msub><mrow><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>≤</mo><mi>n</mi></math></span>, then <em>π</em> is <em>d</em>-separated if <span><math><msub><mrow><mi>i</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>−</mo><msub><mrow><mi>i</mi></mrow><mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>></mo><mi>d</mi></math></span> for each <em>j</em>, <span><math><mn>1</mn><mo><</mo><mi>j</mi><mo>≤</mo><mi>k</mi></math></span>. In this paper we study different enumerative properties on <em>d</em>-separated permutations. We first give a recurrence formula of the numbers <span><math><msup><mrow><mi>c</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span> of <em>d</em>-separated permutations in <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> with exactly <em>k</em> left-to-right minima. Then we study the inversion and co-inversion polynomials of <em>d</em>-separated permutations, and give <em>q</em>-analogue <span><math><msubsup><mrow><mi>c</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>d</mi></mrow></msubsup><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span> and <span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span>-analogue <span><math><msubsup><mrow><mi>c</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow><mrow><mi>d</mi></mrow></msubsup><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span> of <span><math><msup><mrow><mi>c</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span> for any <em>d</em>. Note that when <span><math><mi>d</mi><mo>=</mo><mn>0</mn></math></span>, 0-separated permutations are just all permutations in <span><math><msub><mrow><mi>S</mi></mrow","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102976"},"PeriodicalIF":1.3,"publicationDate":"2025-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145220933","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":"q-Binomial identities finder","authors":"Hao Zhong, Leqi Zhao","doi":"10.1016/j.aam.2025.102965","DOIUrl":"10.1016/j.aam.2025.102965","url":null,"abstract":"<div><div>This paper presents a symbolic computation method for automatically transforming <em>q</em>-hypergeometric identities to <em>q</em>-binomial identities. Through this method, many previously proven <em>q</em>-binomial identities, including <em>q</em>-Saalschütz's formula and <em>q</em>-Suranyi's formula, are re-fund, and numerous new ones are discovered. Moreover, the generation of the identities is accompanied by the corresponding proofs. During the transformation process, different ranges of variable values and various combinations of <em>q</em>-Pochhammer symbols yield different identities. The algorithm maps variable constraints to positive elements in an ordered vector space and employs a backtracking method to provide the feasible variable constraints and <em>q</em>-binomial coefficient combinations for each step.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"172 ","pages":"Article 102965"},"PeriodicalIF":1.3,"publicationDate":"2025-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145026317","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":"A bi-Stirling-Euler-Mahonian polynomial","authors":"Chao Xu, Jiang Zeng","doi":"10.1016/j.aam.2025.102975","DOIUrl":"10.1016/j.aam.2025.102975","url":null,"abstract":"<div><div>Motivated by recent work on (re)mixed Eulerian numbers, we provide a combinatorial interpretation of a subfamily of the remixed Eulerian numbers introduced by Nadeau and Tewari. More specifically, we show that these numbers can be realized as the generating polynomials of permutations with respect to the statistics of left-to-right minima, right-to-left minima, descents, and the mixed major index. Our results generalize both the bi-Stirling-Eulerian polynomials of Carlitz-Scoville and the Stirling-Euler-Mahonian polynomials of Butler.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102975"},"PeriodicalIF":1.3,"publicationDate":"2025-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145027769","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":"On the records and zeros of a deterministic random walk","authors":"Henk Bruin, Robbert Fokkink","doi":"10.1016/j.aam.2025.102963","DOIUrl":"10.1016/j.aam.2025.102963","url":null,"abstract":"<div><div>We settle two questions on sequence A120243 in the OEIS that were raised by Clark Kimberling and partly solve a conjecture of Van de Lune and Arias de Reyna. We extend Kimberling's questions to the framework of deterministic random walks, automatic sequences, and linear recurrences. Our results indicate that there may be a deeper connection between these structures. In particular, we conjecture that the records of deterministic random walks are <em>ξ</em>-Ostrowski automatic for quadratic rotation numbers <em>ξ</em>.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"172 ","pages":"Article 102963"},"PeriodicalIF":1.3,"publicationDate":"2025-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145019453","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 syntactic monoid of θ-free palindromic words","authors":"Kalpana Mahalingam","doi":"10.1016/j.aam.2025.102964","DOIUrl":"10.1016/j.aam.2025.102964","url":null,"abstract":"<div><div>In this paper, we consider words that are free of special type of palindromes, inspired by Watson-Crick complementarity in DNA sequences which we call Watson-Crick palindromic free words or <em>θ</em>-palindromic free words, when <em>θ</em> is an antimorphic involution that incorporates the notion of Watson-Crick complementarity of DNA sequences. We discuss certain algebraic properties of the set of all words that avoids <em>θ</em>-palindromes. We give a complete characterization of the syntactic monoid of the language consisting of all <em>θ</em>-palindromic free words over a given alphabet. We also study the ideals of the corresponding syntactic semigroup obtained using Green's relations. Our results also hold for the more general case, where the Watson-Crick complementarity function is replaced by an arbitrary antimorphic involution.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"172 ","pages":"Article 102964"},"PeriodicalIF":1.3,"publicationDate":"2025-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145019454","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":"On fractal patterns in Ulam words","authors":"Andrei Mandelshtam","doi":"10.1016/j.aam.2025.102954","DOIUrl":"10.1016/j.aam.2025.102954","url":null,"abstract":"<div><div>Ulam words are binary words defined recursively as follows: the length-1 Ulam words are 0 and 1, and a binary word of length <em>n</em> is Ulam if and only if it is expressible uniquely as a concatenation of two shorter, distinct Ulam words. We discover, fully describe, and prove a surprisingly rich structure already in the set of Ulam words containing exactly two 1's. In particular, this leads to a complete description of such words and a logarithmic-time algorithm to determine whether a binary word with two 1's is Ulam. Along the way, we uncover delicate parity and biperiodicity properties, as well as sharp bounds on the number of 0's outside the two 1's. We also show that sets of Ulam words indexed by the number <em>y</em> of 0's between the two 1's have intricate tensor-based hierarchical structures determined by the arithmetic properties of <em>y</em>. This allows us to construct an infinite family of self-similar Ulam-word-based fractals indexed by the set of 2-adic integers, containing the outward Sierpinski gasket as a special case.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"172 ","pages":"Article 102954"},"PeriodicalIF":1.3,"publicationDate":"2025-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145010477","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":"An affine isoperimetric inequality for log-concave functions","authors":"Zengle Zhang , Jiazu Zhou","doi":"10.1016/j.aam.2025.102956","DOIUrl":"10.1016/j.aam.2025.102956","url":null,"abstract":"<div><div>The authors gave an affine isoperimetric inequality <span><span>[41]</span></span> that gives a lower bound for the volume of a polar body and the equality holds if and only if the body is a simplex. In this paper, we give a functional isoperimetric inequality for log-concave functions that contains the affine isoperimetric inequality of Lutwak, Yang and Zhang in <span><span>[41]</span></span>.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"172 ","pages":"Article 102956"},"PeriodicalIF":1.3,"publicationDate":"2025-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144911887","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":"Representation theory and central limit theorems for traces of commutators for compact Lie groups","authors":"Jason Fulman","doi":"10.1016/j.aam.2025.102962","DOIUrl":"10.1016/j.aam.2025.102962","url":null,"abstract":"<div><div>There has been some work in the literature on limit theorems for the trace of commutators for compact Lie groups. We revisit this from the perspective of combinatorial representation theory.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"172 ","pages":"Article 102962"},"PeriodicalIF":1.3,"publicationDate":"2025-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144889329","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":"Counting the number of group orbits by marrying the Burnside process with importance sampling","authors":"Persi Diaconis , Chenyang Zhong","doi":"10.1016/j.aam.2025.102955","DOIUrl":"10.1016/j.aam.2025.102955","url":null,"abstract":"<div><div>This paper introduces a novel and general algorithm for approximately counting the number of orbits under group actions. The method is based on combining the Burnside process and importance sampling. Specializing to unitriangular groups yields an efficient algorithm for estimating the number of conjugacy classes of such groups.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"172 ","pages":"Article 102955"},"PeriodicalIF":1.3,"publicationDate":"2025-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144886384","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":"SL(n) covariant matrix-valued valuations on Orlicz spaces","authors":"Chunna Zeng , Yu Lan","doi":"10.1016/j.aam.2025.102960","DOIUrl":"10.1016/j.aam.2025.102960","url":null,"abstract":"<div><div>All continuous, <span><math><mrow><mi>SL</mi></mrow><mo>(</mo><mi>n</mi><mo>)</mo></math></span> covariant valuations on Orlicz spaces are completely classified without any symmetric assumptions. It is shown that the moment matrix is the only such valuation if <span><math><mi>n</mi><mo>≥</mo><mn>3</mn></math></span>, while a new functional shows up in dimension two.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"172 ","pages":"Article 102960"},"PeriodicalIF":1.3,"publicationDate":"2025-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144886379","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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