Pub Date : 2026-02-01Epub 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":"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}
Pub Date : 2026-02-01Epub Date: 2025-11-06DOI: 10.1016/j.aam.2025.102994
Wei Wang , Da Zhao
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.
{"title":"Graph isomorphism and multivariate graph spectrum","authors":"Wei Wang , 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}
Pub Date : 2026-02-01Epub Date: 2025-11-15DOI: 10.1016/j.aam.2025.102999
Moussa Ahmia , José L. Ramírez , Diego Villamizar
Arslan, Altoum, and Zaarour introduced an inversion statistic for generalized symmetric groups [5]. 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.
{"title":"Inversions in colored permutations, derangements, and involutions","authors":"Moussa Ahmia , José L. Ramírez , 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}
Pub Date : 2026-02-01Epub Date: 2025-10-31DOI: 10.1016/j.aam.2025.102993
Runqiao Li , Andrew Y.Z. Wang , Zheng Xu
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.
{"title":"Block index and integer partitions","authors":"Runqiao Li , Andrew Y.Z. Wang , 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}
Pub Date : 2026-02-01Epub 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":"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}
Pub Date : 2026-02-01Epub Date: 2025-11-10DOI: 10.1016/j.aam.2025.102995
Eri Matsudo , Kanako Oshiro , Gaishi Yamagishi
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 p and any Dehn p-colorable knot K, the minimum number of colors for K is at least . Moreover, we will define the -palette graph for a set of colors. The -palette graphs are quite useful to give candidates of sets of colors which might realize a nontrivially Dehn p-colored diagram. In Appendix, we also prove that for Dehn 5-colorable knot, the minimum number of colors is 4.
{"title":"Minimum numbers of Dehn colors of knots and R-palette graphs","authors":"Eri Matsudo , Kanako Oshiro , 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}
Pub Date : 2026-02-01Epub 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":"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}
Pub Date : 2026-02-01Epub 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":"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}
Pub Date : 2026-02-01Epub 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":"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}
Pub Date : 2026-02-01Epub Date: 2025-11-12DOI: 10.1016/j.aam.2025.102998
Tomasz Jędrzejak
We give a description of the structure of factor rings for the where n is an integer (which is not a square). For example, we prove that is isomorphic to the ring of integers modulo for relatively prime . We also characterize the structure of for arbitrary integers . Finally, we describe for non-principal ideals I. We also present many corollaries regarding irreducible and prime elements in and give numerous examples. We only use methods from elementary number theory and basic ring theory.
{"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}
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