Pub Date : 2025-10-08DOI: 10.1016/j.aam.2025.102979
Yihan Xiao , Rui Xiong , Haofeng Zhang
We develop a family of new combinatorial models for key polynomials. It is similar to the hybrid pipe dream model for Schubert polynomials defined recently by Knutson and Udell.
{"title":"Hybrid pipe dreams for key polynomials","authors":"Yihan Xiao , Rui Xiong , Haofeng Zhang","doi":"10.1016/j.aam.2025.102979","DOIUrl":"10.1016/j.aam.2025.102979","url":null,"abstract":"<div><div>We develop a family of new combinatorial models for key polynomials. It is similar to the hybrid pipe dream model for Schubert polynomials defined recently by Knutson and Udell.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102979"},"PeriodicalIF":1.3,"publicationDate":"2025-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145268470","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 : 2025-10-07DOI: 10.1016/j.aam.2025.102981
George E. Andrews , Mohamed El Bachraoui
We focus on certain integer partitions and their weighted analogues with conditions on the interval of their parts. The q-double series turn out to be more fruitful as generating functions for our sequences. We give explicit formulas for the number of such partitions, we derive identities involving integer partitions, and we prove that some of our weighted sequences are positive. Furthermore, we state two curious conjectures on the coefficients of two q-double series.
{"title":"Formulas and conjectures for partitions with restrictions on interval of parts","authors":"George E. Andrews , Mohamed El Bachraoui","doi":"10.1016/j.aam.2025.102981","DOIUrl":"10.1016/j.aam.2025.102981","url":null,"abstract":"<div><div>We focus on certain integer partitions and their weighted analogues with conditions on the interval of their parts. The <em>q</em>-double series turn out to be more fruitful as generating functions for our sequences. We give explicit formulas for the number of such partitions, we derive identities involving integer partitions, and we prove that some of our weighted sequences are positive. Furthermore, we state two curious conjectures on the coefficients of two <em>q</em>-double series.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102981"},"PeriodicalIF":1.3,"publicationDate":"2025-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145268596","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 : 2025-10-06DOI: 10.1016/j.aam.2025.102978
Lu Zhang
In this paper, we consider a class of generalized chord integrals in integral geometry, where the integrand is a generalized kernel that replaces the Riesz kernel. The generalized chord measure arises from the study of the generalized chord integral of convex bodies. We pose the Minkowski problem for the generalized chord measure and obtain the existence of solutions to the related Minkowski problem.
{"title":"A class of generalized chord Minkowski problems","authors":"Lu Zhang","doi":"10.1016/j.aam.2025.102978","DOIUrl":"10.1016/j.aam.2025.102978","url":null,"abstract":"<div><div>In this paper, we consider a class of generalized chord integrals in integral geometry, where the integrand is a generalized kernel that replaces the Riesz kernel. The generalized chord measure arises from the study of the generalized chord integral of convex bodies. We pose the Minkowski problem for the generalized chord measure and obtain the existence of solutions to the related Minkowski problem.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102978"},"PeriodicalIF":1.3,"publicationDate":"2025-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145268595","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 : 2025-09-30DOI: 10.1016/j.aam.2025.102976
Rosena R.X. Du, Yun Li
Let d be a nonnegative integer, a d-separated permutation is a permutation in which every two left-to-right minima are at distance greater than d. More precisely, for , suppose that are the left-to-right minima of π with and , then π is d-separated if for each j, . In this paper we study different enumerative properties on d-separated permutations. We first give a recurrence formula of the numbers of d-separated permutations in with exactly k left-to-right minima. Then we study the inversion and co-inversion polynomials of d-separated permutations, and give q-analogue and -analogue of for any d. Note that when , 0-separated permutations are just all permutations in
{"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}
Pub Date : 2025-09-10DOI: 10.1016/j.aam.2025.102965
Hao Zhong, Leqi Zhao
This paper presents a symbolic computation method for automatically transforming q-hypergeometric identities to q-binomial identities. Through this method, many previously proven q-binomial identities, including q-Saalschütz's formula and q-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 q-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 q-binomial coefficient combinations for each step.
{"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}
Pub Date : 2025-09-09DOI: 10.1016/j.aam.2025.102975
Chao Xu, Jiang Zeng
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.
{"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}
Pub Date : 2025-09-09DOI: 10.1016/j.aam.2025.102963
Henk Bruin, Robbert Fokkink
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 ξ-Ostrowski automatic for quadratic rotation numbers ξ.
我们解决了Clark Kimberling提出的关于OEIS中A120243序列的两个问题，部分解决了Van de Lune和Arias de Reyna的猜想。我们将金伯林的问题扩展到确定性随机漫步、自动序列和线性递归的框架。我们的研究结果表明，这些结构之间可能存在更深层次的联系。特别地，我们推测确定性随机漫步的记录对于二次旋转数ξ是ξ- ostrowski自动的。
{"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}
Pub Date : 2025-09-09DOI: 10.1016/j.aam.2025.102964
Kalpana Mahalingam
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 θ-palindromic free words, when θ 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 θ-palindromes. We give a complete characterization of the syntactic monoid of the language consisting of all θ-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.
{"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}
Pub Date : 2025-09-08DOI: 10.1016/j.aam.2025.102954
Andrei Mandelshtam
Ulam words are binary words defined recursively as follows: the length-1 Ulam words are 0 and 1, and a binary word of length n 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 y of 0's between the two 1's have intricate tensor-based hierarchical structures determined by the arithmetic properties of y. 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.
{"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}
Pub Date : 2025-08-29DOI: 10.1016/j.aam.2025.102956
Zengle Zhang , Jiazu Zhou
The authors gave an affine isoperimetric inequality [41] 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 [41].
{"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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：