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Formulas and conjectures for partitions with restrictions on interval of parts 部分间隔有限制的分区的公式和猜想
IF 1.3 3区 数学
Advances in Applied Mathematics Pub Date : 2026-02-01 Epub Date: 2025-10-07 DOI: 10.1016/j.aam.2025.102981
George E. Andrews , Mohamed El Bachraoui
{"title":"Formulas and conjectures for partitions with restrictions on interval of parts","authors":"George E. Andrews ,&nbsp;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":"2026-02-01","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}
引用次数: 0
Graph isomorphism and multivariate graph spectrum 图同构与多元图谱
IF 1.3 3区 数学
Advances in Applied Mathematics Pub Date : 2026-02-01 Epub Date: 2025-11-06 DOI: 10.1016/j.aam.2025.102994
Wei Wang , Da Zhao
{"title":"Graph isomorphism and multivariate graph spectrum","authors":"Wei Wang ,&nbsp;Da Zhao","doi":"10.1016/j.aam.2025.102994","DOIUrl":"10.1016/j.aam.2025.102994","url":null,"abstract":"<div><div>We provide a criterion to show that a graph is identified by its multivariate graph spectrum. Haemers conjectured that almost all graphs are identified by their spectra. Our approach suggests that almost all graphs are identified by their generalized block Laplacian spectra.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102994"},"PeriodicalIF":1.3,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145474011","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}
引用次数: 0
Inversions in colored permutations, derangements, and involutions 彩色排列、无序和内联的反转
IF 1.3 3区 数学
Advances in Applied Mathematics Pub Date : 2026-02-01 Epub Date: 2025-11-15 DOI: 10.1016/j.aam.2025.102999
Moussa Ahmia , José L. Ramírez , Diego Villamizar
{"title":"Inversions in colored permutations, derangements, and involutions","authors":"Moussa Ahmia ,&nbsp;José L. Ramírez ,&nbsp;Diego Villamizar","doi":"10.1016/j.aam.2025.102999","DOIUrl":"10.1016/j.aam.2025.102999","url":null,"abstract":"<div><div>Arslan, Altoum, and Zaarour introduced an inversion statistic for generalized symmetric groups <span><span>[5]</span></span>. In this work, we study the distribution of this statistic over colored permutations, including derangements and involutions. By establishing a bijective correspondence between colored permutations and colored Lehmer codes, we develop a unified framework for enumerating colored Mahonian numbers and analyzing their combinatorial properties. We derive explicit formulas, recurrence relations, and generating functions for the number of inversions in these families, extending classical results to the colored setting. We conclude with explicit expressions for inversions in colored derangements and involutions.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102999"},"PeriodicalIF":1.3,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145528747","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}
引用次数: 0
Block index and integer partitions 块索引和整数分区
IF 1.3 3区 数学
Advances in Applied Mathematics Pub Date : 2026-02-01 Epub Date: 2025-10-31 DOI: 10.1016/j.aam.2025.102993
Runqiao Li , Andrew Y.Z. Wang , Zheng Xu
{"title":"Block index and integer partitions","authors":"Runqiao Li ,&nbsp;Andrew Y.Z. Wang ,&nbsp;Zheng Xu","doi":"10.1016/j.aam.2025.102993","DOIUrl":"10.1016/j.aam.2025.102993","url":null,"abstract":"<div><div>In this work, we introduce a new partition statistic, named block index, and explore its relationship with other well-known statistics, including Dyson's crank. We delve into the combinatorial significance of the block index, shedding light on its role in revealing the more intricate structure of certain recently discovered partition identities.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102993"},"PeriodicalIF":1.3,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145424662","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}
引用次数: 0
Enumerative proof of a curious congruence for Eulerian numbers 欧拉数的一个奇异同余的枚举证明
IF 1.3 3区 数学
Advances in Applied Mathematics Pub Date : 2026-02-01 Epub Date: 2025-10-09 DOI: 10.1016/j.aam.2025.102977
Xiangzi Meng , Hao Pan
{"title":"Enumerative proof of a curious congruence for Eulerian numbers","authors":"Xiangzi Meng ,&nbsp;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>&lt;</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":"2026-02-01","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}
引用次数: 0
Minimum numbers of Dehn colors of knots and R-palette graphs 结点和r -调色板图的Dehn颜色的最小数目
IF 1.3 3区 数学
Advances in Applied Mathematics Pub Date : 2026-02-01 Epub Date: 2025-11-10 DOI: 10.1016/j.aam.2025.102995
Eri Matsudo , Kanako Oshiro , Gaishi Yamagishi
{"title":"Minimum numbers of Dehn colors of knots and R-palette graphs","authors":"Eri Matsudo ,&nbsp;Kanako Oshiro ,&nbsp;Gaishi Yamagishi","doi":"10.1016/j.aam.2025.102995","DOIUrl":"10.1016/j.aam.2025.102995","url":null,"abstract":"<div><div>This is the first paper which discusses minimum numbers of “region” colors for knots, while minimum numbers of arc colors are well-studied. In this paper, we consider minimum numbers of colors of knots for Dehn colorings. In particular, we will show that for any odd prime number <em>p</em> and any Dehn <em>p</em>-colorable knot <em>K</em>, the minimum number of colors for <em>K</em> is at least <span><math><mo>⌊</mo><msub><mrow><mi>log</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⁡</mo><mi>p</mi><mo>⌋</mo><mo>+</mo><mn>2</mn></math></span>. Moreover, we will define the <span><math><mi>R</mi></math></span>-palette graph for a set of colors. The <span><math><mi>R</mi></math></span>-palette graphs are quite useful to give candidates of sets of colors which might realize a nontrivially Dehn <em>p</em>-colored diagram. In Appendix, we also prove that for Dehn 5-colorable knot, the minimum number of colors is 4.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102995"},"PeriodicalIF":1.3,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145528744","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}
引用次数: 0
A study on T-equivalent graphs 关于t -等价图的研究
IF 1.3 3区 数学
Advances in Applied Mathematics Pub Date : 2026-02-01 Epub Date: 2025-10-20 DOI: 10.1016/j.aam.2025.102985
Fengming Dong , Meiqiao Zhang
{"title":"A study on T-equivalent graphs","authors":"Fengming Dong ,&nbsp;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":"2026-02-01","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}
引用次数: 0
On the enumeration of double cosets and self-inverse double cosets 关于双陪集和自逆双陪集的枚举
IF 1.3 3区 数学
Advances in Applied Mathematics Pub Date : 2026-02-01 Epub Date: 2025-10-13 DOI: 10.1016/j.aam.2025.102982
Ludovic Schwob
{"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":"2026-02-01","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}
引用次数: 0
A bi-Stirling-Euler-Mahonian polynomial 一个双stirling - euler - mahonian多项式
IF 1.3 3区 数学
Advances in Applied Mathematics Pub Date : 2026-02-01 Epub Date: 2025-09-09 DOI: 10.1016/j.aam.2025.102975
Chao Xu, Jiang Zeng
{"title":"A bi-Stirling-Euler-Mahonian polynomial","authors":"Chao Xu,&nbsp;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":"2026-02-01","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}
引用次数: 0
The structure of factor rings of Z[n] Z[n]因子环的结构
IF 1.3 3区 数学
Advances in Applied Mathematics Pub Date : 2026-02-01 Epub Date: 2025-11-12 DOI: 10.1016/j.aam.2025.102998
Tomasz Jędrzejak
{"title":"The structure of factor rings of Z[n]","authors":"Tomasz Jędrzejak","doi":"10.1016/j.aam.2025.102998","DOIUrl":"10.1016/j.aam.2025.102998","url":null,"abstract":"<div><div>We give a description of the structure of factor rings for the <span><math><mi>Z</mi><mrow><mo>[</mo><msqrt><mrow><mi>n</mi></mrow></msqrt><mo>]</mo></mrow></math></span> where <em>n</em> is an integer (which is not a square). For example, we prove that <span><math><mi>Z</mi><mrow><mo>[</mo><msqrt><mrow><mi>n</mi></mrow></msqrt><mo>]</mo></mrow><mo>/</mo><mrow><mo>(</mo><mi>a</mi><mo>+</mo><mi>b</mi><msqrt><mrow><mi>n</mi></mrow></msqrt><mo>)</mo></mrow></math></span> is isomorphic to the ring of integers modulo <span><math><mo>|</mo><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mi>n</mi><msup><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>|</mo></math></span> for relatively prime <span><math><mi>a</mi><mo>,</mo><mi>b</mi></math></span>. We also characterize the structure of <span><math><mi>Z</mi><mrow><mo>[</mo><msqrt><mrow><mi>n</mi></mrow></msqrt><mo>]</mo></mrow><mo>/</mo><mrow><mo>(</mo><mi>a</mi><mo>+</mo><mi>b</mi><msqrt><mrow><mi>n</mi></mrow></msqrt><mo>)</mo></mrow></math></span> for arbitrary integers <span><math><mi>a</mi><mo>,</mo><mi>b</mi></math></span>. Finally, we describe <span><math><mi>Z</mi><mrow><mo>[</mo><msqrt><mrow><mi>n</mi></mrow></msqrt><mo>]</mo></mrow><mo>/</mo><mi>I</mi></math></span> for non-principal ideals <em>I</em>. We also present many corollaries regarding irreducible and prime elements in <span><math><mi>Z</mi><mrow><mo>[</mo><msqrt><mrow><mi>n</mi></mrow></msqrt><mo>]</mo></mrow></math></span> and give numerous examples. We only use methods from elementary number theory and basic ring theory.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102998"},"PeriodicalIF":1.3,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145528745","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}
引用次数: 0
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