Pub Date : 2025-10-20DOI: 10.1016/j.aam.2025.102985
Fengming Dong , Meiqiao Zhang
In his article [J. Comb. Theory Ser. B16 (1974), 168–174], Tutte called two graphs T-equivalent (i.e., codichromatic) if they have the same Tutte polynomial and showed that graphs G and are T-equivalent if is obtained from G by flipping a rotor (i.e., replacing it by its mirror) of order at most 5, where a rotor of order k in G is an induced subgraph R having an automorphism ψ with a vertex orbit of size k such that every vertex of R is only adjacent to vertices in R unless it is in this vertex orbit. In this article, we show the above result due to Tutte can be extended to a rotor R of order if the subgraph of G induced by all those edges of G which are not in R satisfies certain conditions. Also, we provide a new method for generating infinitely many non-isomorphic T-equivalent pairs of graphs.
在他的文章中[J]。合成杆。Ser的理论。B 16 (1974), 168 - 174], Tutte叫两个图形T-equivalent(即codichromatic)如果他们有相同的Tutte多项式和显示,图G, G T-equivalent如果G是来自G翻转一个转子(即取代它的镜像)的订单最多5 k阶转子在G是一种诱导子图R有自同构与一个顶点ψ轨道{ψ(u):我≥0}的k大小的每个顶点只相邻顶点在R,除非它是在这个顶点轨道。在本文中，我们证明了由于Tutte的上述结果可以推广到k≥6阶的转子R，如果G的所有不在R中的边所诱导的G的子图满足一定的条件。此外，我们还提供了一种生成无限多个非同构t等价图对的新方法。
{"title":"A study on T-equivalent graphs","authors":"Fengming Dong , Meiqiao Zhang","doi":"10.1016/j.aam.2025.102985","DOIUrl":"10.1016/j.aam.2025.102985","url":null,"abstract":"<div><div>In his article [<em>J. Comb. Theory Ser. B</em> <strong>16</strong> (1974), 168–174], Tutte called two graphs <em>T</em>-equivalent (i.e., codichromatic) if they have the same Tutte polynomial and showed that graphs <em>G</em> and <span><math><msup><mrow><mi>G</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> are <em>T</em>-equivalent if <span><math><msup><mrow><mi>G</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> is obtained from <em>G</em> by flipping a rotor (i.e., replacing it by its mirror) of order at most 5, where a rotor of order <em>k</em> in <em>G</em> is an induced subgraph <em>R</em> having an automorphism <em>ψ</em> with a vertex orbit <span><math><mo>{</mo><msup><mrow><mi>ψ</mi></mrow><mrow><mi>i</mi></mrow></msup><mo>(</mo><mi>u</mi><mo>)</mo><mo>:</mo><mi>i</mi><mo>≥</mo><mn>0</mn><mo>}</mo></math></span> of size <em>k</em> such that every vertex of <em>R</em> is only adjacent to vertices in <em>R</em> unless it is in this vertex orbit. In this article, we show the above result due to Tutte can be extended to a rotor <em>R</em> of order <span><math><mi>k</mi><mo>≥</mo><mn>6</mn></math></span> if the subgraph of <em>G</em> induced by all those edges of <em>G</em> which are not in <em>R</em> satisfies certain conditions. Also, we provide a new method for generating infinitely many non-isomorphic <em>T</em>-equivalent pairs of graphs.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102985"},"PeriodicalIF":1.3,"publicationDate":"2025-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145363042","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 : 2025-10-13DOI: 10.1016/j.aam.2025.102982
Ludovic Schwob
Double cosets appear in many contexts in combinatorics, for example in the enumeration of certain objects up to symmetries. Double cosets in a quotient of the form have an inverse, and can be their own inverse. In this paper we present various formulas enumerating double cosets, and in particular self-inverse double cosets. We study double cosets in classical groups, especially the symmetric groups and the general linear groups, explaining how to obtain the information on their conjugacy classes required to apply our formulas. We also consider double cosets of parabolic subgroups of Coxeter groups of type B.
{"title":"On the enumeration of double cosets and self-inverse double cosets","authors":"Ludovic Schwob","doi":"10.1016/j.aam.2025.102982","DOIUrl":"10.1016/j.aam.2025.102982","url":null,"abstract":"<div><div>Double cosets appear in many contexts in combinatorics, for example in the enumeration of certain objects up to symmetries. Double cosets in a quotient of the form <span><math><mi>H</mi><mo>﹨</mo><mi>G</mi><mo>/</mo><mi>H</mi></math></span> have an inverse, and can be their own inverse. In this paper we present various formulas enumerating double cosets, and in particular self-inverse double cosets. We study double cosets in classical groups, especially the symmetric groups and the general linear groups, explaining how to obtain the information on their conjugacy classes required to apply our formulas. We also consider double cosets of parabolic subgroups of Coxeter groups of type B.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102982"},"PeriodicalIF":1.3,"publicationDate":"2025-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145320793","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 : 2025-10-10DOI: 10.1016/j.aam.2025.102983
Christine Cho , James Oxley , Suijie Wang
If and are circuits in a matroid M with in and e in , then M has a circuit such that . This strong circuit elimination axiom is inherently asymmetric. A matroid M has the symmetric strong circuit elimination property (SSCE) if, when the above conditions hold and , there is a circuit with . We prove that a connected matroid has this property if and only if it has no two skew circuits. We also characterize such matroids in terms of forbidden series minors, and we give a new matroid axiom system that is built around a modification of SSCE.
{"title":"The symmetric strong circuit elimination property","authors":"Christine Cho , James Oxley , Suijie Wang","doi":"10.1016/j.aam.2025.102983","DOIUrl":"10.1016/j.aam.2025.102983","url":null,"abstract":"<div><div>If <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> are circuits in a matroid <em>M</em> with <span><math><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> in <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>−</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and <em>e</em> in <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∩</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, then <em>M</em> has a circuit <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> such that <span><math><mi>e</mi><mo>∈</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>⊆</mo><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∪</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>−</mo><mi>e</mi></math></span>. This strong circuit elimination axiom is inherently asymmetric. A matroid <em>M</em> has the symmetric strong circuit elimination property (SSCE) if, when the above conditions hold and <span><math><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∈</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>−</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>, there is a circuit <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow><mrow><mo>′</mo></mrow></msubsup></math></span> with <span><math><mo>{</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>}</mo><mo>⊆</mo><msubsup><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow><mrow><mo>′</mo></mrow></msubsup><mo>⊆</mo><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∪</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>−</mo><mi>e</mi></math></span>. We prove that a connected matroid has this property if and only if it has no two skew circuits. We also characterize such matroids in terms of forbidden series minors, and we give a new matroid axiom system that is built around a modification of SSCE.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102983"},"PeriodicalIF":1.3,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145268469","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 : 2025-10-09DOI: 10.1016/j.aam.2025.102977
Xiangzi Meng , Hao Pan
The Eulerian number counts all permutations on having exactly k ascents. In this paper, we give an enumerative proof of the following congruence: where p is prime, and .
欧拉数< nk >计算{0,1，…，n−1}上的所有恰好有k个上升的排列。本文给出了下列同余的一个枚举证明：< ap−1bp+l >≡(−1)b(l+1)a−1(a−1b)(modp)，其中p为素数，0≤b<；a且0≤l≤p−1。
{"title":"Enumerative proof of a curious congruence for Eulerian numbers","authors":"Xiangzi Meng , Hao Pan","doi":"10.1016/j.aam.2025.102977","DOIUrl":"10.1016/j.aam.2025.102977","url":null,"abstract":"<div><div>The Eulerian number <span><math><mo>〈</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>〉</mo></math></span> counts all permutations on <span><math><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>}</mo></math></span> having exactly <em>k</em> ascents. In this paper, we give an enumerative proof of the following congruence:<span><span><span><math><mrow><mo>〈</mo><mtable><mtr><mtd><mrow><mi>a</mi><mi>p</mi><mo>−</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>b</mi><mi>p</mi><mo>+</mo><mi>l</mi></mrow></mtd></mtr></mtable><mo>〉</mo></mrow><mo>≡</mo><msup><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>b</mi></mrow></msup><msup><mrow><mo>(</mo><mi>l</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>a</mi><mo>−</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>a</mi><mo>−</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mi>b</mi></mtd></mtr></mtable><mo>)</mo></mrow><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mi>p</mi><mo>)</mo><mo>,</mo></math></span></span></span> where <em>p</em> is prime, <span><math><mn>0</mn><mo>≤</mo><mi>b</mi><mo><</mo><mi>a</mi></math></span> and <span><math><mn>0</mn><mo>≤</mo><mi>l</mi><mo>≤</mo><mi>p</mi><mo>−</mo><mn>1</mn></math></span>.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102977"},"PeriodicalIF":1.3,"publicationDate":"2025-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145268467","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 : 2025-10-09DOI: 10.1016/j.aam.2025.102980
Hongying Lin , Bo Zhou
Let m be a positive integer. Brualdi and Hoffman proposed the problem to determine the (connected) graphs with maximum adjacency spectral radius in a given graph class and they posed a conjecture for the class of graphs with given size m. After partial results due to Friedland and Stanley, Rowlinson completely confirmed the conjecture. The distance spectral radius of a connected graph is the largest eigenvalue of its distance matrix. We investigate the problem to determine the connected graphs with minimum distance spectral radius in the class of graphs with size m. Given m, there is exactly one positive integer n such that . We establish some structural properties of the extremal graphs for all m and solve the problem for . We give a conjecture for the remaining case. To prove the main results, we also determine the complements of forests of fixed order with large and small distance spectral radius.
{"title":"Extremal distance spectral radius of graphs with fixed size","authors":"Hongying Lin , Bo Zhou","doi":"10.1016/j.aam.2025.102980","DOIUrl":"10.1016/j.aam.2025.102980","url":null,"abstract":"<div><div>Let <em>m</em> be a positive integer. Brualdi and Hoffman proposed the problem to determine the (connected) graphs with maximum adjacency spectral radius in a given graph class and they posed a conjecture for the class of graphs with given size <em>m</em>. After partial results due to Friedland and Stanley, Rowlinson completely confirmed the conjecture. The distance spectral radius of a connected graph is the largest eigenvalue of its distance matrix. We investigate the problem to determine the connected graphs with minimum distance spectral radius in the class of graphs with size <em>m</em>. Given <em>m</em>, there is exactly one positive integer <em>n</em> such that <span><math><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow><mo><</mo><mi>m</mi><mo>≤</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow></math></span>. We establish some structural properties of the extremal graphs for all <em>m</em> and solve the problem for <span><math><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow><mo>+</mo><mi>max</mi><mo></mo><mo>{</mo><mfrac><mrow><mi>n</mi><mo>−</mo><mn>6</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mn>1</mn><mo>}</mo><mo>≤</mo><mi>m</mi><mo>≤</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow></math></span>. We give a conjecture for the remaining case. To prove the main results, we also determine the complements of forests of fixed order with large and small distance spectral radius.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102980"},"PeriodicalIF":1.3,"publicationDate":"2025-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145268468","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 : 2025-10-08DOI: 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}
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