Pub Date : 2025-11-24DOI: 10.1016/j.aam.2025.103001
Takuya Saito
Discriminantal arrangements are hyperplane arrangements that are generalization of braid arrangements. They are constructed from given hyperplane arrangements, but their combinatorics are not invariant under combinatorial equivalence. However, it is known that the combinatorics of the discriminantal arrangements are constant on a Zariski open set of the space of hyperplane arrangements. In the present paper, we introduce -singularity varieties in the space of hyperplane arrangements to classify discriminantal arrangements and show that the Zariski open set is the complement of -singularity varieties. We study their basic properties and operations and provide examples, including infinite families of -singularity varieties. In particular, the operation that we call degeneration is a powerful tool for constructing -singularity varieties. As an application, we provide a list of -singularity varieties for spaces of small line arrangements.
{"title":"Degeneration in discriminantal arrangements","authors":"Takuya Saito","doi":"10.1016/j.aam.2025.103001","DOIUrl":"10.1016/j.aam.2025.103001","url":null,"abstract":"<div><div>Discriminantal arrangements are hyperplane arrangements that are generalization of braid arrangements. They are constructed from given hyperplane arrangements, but their combinatorics are not invariant under combinatorial equivalence. However, it is known that the combinatorics of the discriminantal arrangements are constant on a Zariski open set of the space of hyperplane arrangements. In the present paper, we introduce <span><math><mo>(</mo><mi>T</mi><mo>,</mo><mi>r</mi><mo>)</mo></math></span>-singularity varieties in the space of hyperplane arrangements to classify discriminantal arrangements and show that the Zariski open set is the complement of <span><math><mo>(</mo><mi>T</mi><mo>,</mo><mi>r</mi><mo>)</mo></math></span>-singularity varieties. We study their basic properties and operations and provide examples, including infinite families of <span><math><mo>(</mo><mi>T</mi><mo>,</mo><mi>r</mi><mo>)</mo></math></span>-singularity varieties. In particular, the operation that we call degeneration is a powerful tool for constructing <span><math><mo>(</mo><mi>T</mi><mo>,</mo><mi>r</mi><mo>)</mo></math></span>-singularity varieties. As an application, we provide a list of <span><math><mo>(</mo><mi>T</mi><mo>,</mo><mi>r</mi><mo>)</mo></math></span>-singularity varieties for spaces of small line arrangements.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 103001"},"PeriodicalIF":1.3,"publicationDate":"2025-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145623588","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 : 2025-11-18DOI: 10.1016/j.aam.2025.103000
Chang Liu , Jianping Li , Shuchao Li , Yuantian Yu
<div><div>The Brualdi-Hoffman-Turán problem, a central topic in spectral graph theory, seeks to determine maximum spectral radius <span><math><mi>ρ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of an <em>F</em>-free graph <em>G</em> with <em>m</em> edges. This problem has attracted significant attention in recent years. Let <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mi>l</mi></mrow><mrow><mo>+</mo></mrow></msubsup></math></span> denote the theta graph obtained by adding a chord between two vertices at distance 2 on cycle <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>l</mi></mrow></msub></math></span>. Zhai, Lin, and Shu <span><span>[32]</span></span> conjectured that, for <span><math><mi>k</mi><mo>⩾</mo><mn>2</mn></math></span> and sufficiently large <em>m</em>, if <em>G</em> is <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>-free or <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>2</mn></mrow></msub></math></span>-free, then <span><math><mi>ρ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>⩽</mo><mfrac><mrow><mi>k</mi><mo>−</mo><mn>1</mn><mo>+</mo><msqrt><mrow><mn>4</mn><mi>m</mi><mo>−</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>1</mn></mrow></msqrt></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>, with equality if and only if <span><math><mi>G</mi><mo>≅</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>∨</mo><mo>(</mo><mfrac><mrow><mi>m</mi></mrow><mrow><mi>k</mi></mrow></mfrac><mo>−</mo><mfrac><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>. This conjecture was highlighted in Liu and Ning's survey <span><span>[18]</span></span> as one of the twenty unsolved problems in spectral graph theory. Subsequently, Y.T. Li proposed an even stronger conjecture, which claims that the upper bound on <span><math><mi>ρ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and the corresponding extremal graph hold for <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mrow><mo>+</mo></mrow></msubsup></math></span>-free, or <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>2</mn></mrow><mrow><mo>+</mo></mrow></msubsup></math></span>-free graphs. Recently, Li, Zhai, and Shu <span><span>[14]</span></span> resolved both conjectures completely. Note that the above extremal graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>∨</mo><mo>(</mo><mfrac><mrow><mi>m</mi></mrow><mrow><mi>k</mi></mrow></mfrac><mo>−</mo><mfrac><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> is well-defined only if <span><math><mi>m</mi><mo>+</mo><mfrac><mrow><mi>k</mi><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</
{"title":"A Brualdi-Hoffman-Turán problem on theta graph","authors":"Chang Liu , Jianping Li , Shuchao Li , Yuantian Yu","doi":"10.1016/j.aam.2025.103000","DOIUrl":"10.1016/j.aam.2025.103000","url":null,"abstract":"<div><div>The Brualdi-Hoffman-Turán problem, a central topic in spectral graph theory, seeks to determine maximum spectral radius <span><math><mi>ρ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of an <em>F</em>-free graph <em>G</em> with <em>m</em> edges. This problem has attracted significant attention in recent years. Let <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mi>l</mi></mrow><mrow><mo>+</mo></mrow></msubsup></math></span> denote the theta graph obtained by adding a chord between two vertices at distance 2 on cycle <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>l</mi></mrow></msub></math></span>. Zhai, Lin, and Shu <span><span>[32]</span></span> conjectured that, for <span><math><mi>k</mi><mo>⩾</mo><mn>2</mn></math></span> and sufficiently large <em>m</em>, if <em>G</em> is <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>-free or <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>2</mn></mrow></msub></math></span>-free, then <span><math><mi>ρ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>⩽</mo><mfrac><mrow><mi>k</mi><mo>−</mo><mn>1</mn><mo>+</mo><msqrt><mrow><mn>4</mn><mi>m</mi><mo>−</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>1</mn></mrow></msqrt></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>, with equality if and only if <span><math><mi>G</mi><mo>≅</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>∨</mo><mo>(</mo><mfrac><mrow><mi>m</mi></mrow><mrow><mi>k</mi></mrow></mfrac><mo>−</mo><mfrac><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>. This conjecture was highlighted in Liu and Ning's survey <span><span>[18]</span></span> as one of the twenty unsolved problems in spectral graph theory. Subsequently, Y.T. Li proposed an even stronger conjecture, which claims that the upper bound on <span><math><mi>ρ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and the corresponding extremal graph hold for <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mrow><mo>+</mo></mrow></msubsup></math></span>-free, or <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>2</mn></mrow><mrow><mo>+</mo></mrow></msubsup></math></span>-free graphs. Recently, Li, Zhai, and Shu <span><span>[14]</span></span> resolved both conjectures completely. Note that the above extremal graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>∨</mo><mo>(</mo><mfrac><mrow><mi>m</mi></mrow><mrow><mi>k</mi></mrow></mfrac><mo>−</mo><mfrac><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> is well-defined only if <span><math><mi>m</mi><mo>+</mo><mfrac><mrow><mi>k</mi><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 103000"},"PeriodicalIF":1.3,"publicationDate":"2025-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145578779","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 : 2025-11-15DOI: 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":"2025-11-15","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 : 2025-11-15DOI: 10.1016/j.aam.2025.102997
William Gustafson
We introduce posets of simple vertex labeled minors of graphs and a generalization to the level of polymatroids, collectively termed minor posets. We show that any minor poset is isomorphic to the face poset of a regular CW sphere, and in particular, is Eulerian. We establish cd-index inequalities induced by strong maps, a tight upper bound for cd-indices of minor posets and a tight lower bound for cd-indices of minor posets arising from lattices of maximal length.
{"title":"Lattice minors and Eulerian posets","authors":"William Gustafson","doi":"10.1016/j.aam.2025.102997","DOIUrl":"10.1016/j.aam.2025.102997","url":null,"abstract":"<div><div>We introduce posets of simple vertex labeled minors of graphs and a generalization to the level of polymatroids, collectively termed minor posets. We show that any minor poset is isomorphic to the face poset of a regular CW sphere, and in particular, is Eulerian. We establish <strong>cd</strong>-index inequalities induced by strong maps, a tight upper bound for <strong>cd</strong>-indices of minor posets and a tight lower bound for <strong>cd</strong>-indices of minor posets arising from lattices of maximal length.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102997"},"PeriodicalIF":1.3,"publicationDate":"2025-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145528746","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 : 2025-11-12DOI: 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":"2025-11-12","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}
Pub 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":"2025-11-10","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 : 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":"2025-11-06","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 : 2025-11-06DOI: 10.1016/j.aam.2025.102996
Mark Dukes, Andrew Mullins
Banach's matchbox problem considers the setting of two matchboxes that each initially contain the same number of matches. Boxes are chosen with equal probability and a match removed each time. The problem concerns the law of the number of matches remaining in one box once the other box empties. Knuth considered a generalization of this problem whereby big-choosers arrive with probability p and remove a match from the box with the most number remaining, and little-choosers arrive with probability and remove a match from the box with the least number remaining.
In this paper we consider Knuth's generalization for the case of k matchboxes in which there are big-choosers and little-choosers. We determine the generating function for the expected number of matches remaining in matchboxes once a box first empties, a quantity we refer to as the ‘residue’. Interestingly, this generating function is a quotient whose denominator contains a generating function for a special case of the Raney numbers. The form for this generating function allows us to give an expression for the expected residue in terms of a sum that involves diagonal state return probabilities, where a diagonal state is a configuration in which all matchboxes each contain the same number of matches. We use analytic techniques to determine the asymptotic behavior of this expected value for all values of p, which involves the study of an asymmetric random walk.
In addition to this we consider the expected value of the order of the first return to a diagonal state and determine the asymptotic behavior of this quantity. The coefficients of the diagonal state probability generating function are shown to be related to ‘manila folder configurations in a filing cabinet’, and we make this connection precise. This allows us to use known results for the enumeration of such manila folder configurations to give a closed form expression for the diagonal state return probabilities.
{"title":"Knuth's big-chooser matchbox process: the case of many matchboxes","authors":"Mark Dukes, Andrew Mullins","doi":"10.1016/j.aam.2025.102996","DOIUrl":"10.1016/j.aam.2025.102996","url":null,"abstract":"<div><div>Banach's matchbox problem considers the setting of two matchboxes that each initially contain the same number of matches. Boxes are chosen with equal probability and a match removed each time. The problem concerns the law of the number of matches remaining in one box once the other box empties. Knuth considered a generalization of this problem whereby <em>big-choosers</em> arrive with probability <em>p</em> and remove a match from the box with the most number remaining, and <em>little-choosers</em> arrive with probability <span><math><mn>1</mn><mo>−</mo><mi>p</mi></math></span> and remove a match from the box with the least number remaining.</div><div>In this paper we consider Knuth's generalization for the case of <em>k</em> matchboxes in which there are <em>big-choosers</em> and <em>little-choosers</em>. We determine the generating function for the expected number of matches remaining in <span><math><mi>k</mi><mo>−</mo><mn>1</mn></math></span> matchboxes once a box first empties, a quantity we refer to as the ‘residue’. Interestingly, this generating function is a quotient whose denominator contains a generating function for a special case of the Raney numbers. The form for this generating function allows us to give an expression for the expected residue in terms of a sum that involves diagonal state return probabilities, where a diagonal state is a configuration in which all matchboxes each contain the same number of matches. We use analytic techniques to determine the asymptotic behavior of this expected value for all values of <em>p</em>, which involves the study of an asymmetric random walk.</div><div>In addition to this we consider the expected value of the order of the first return to a diagonal state and determine the asymptotic behavior of this quantity. The coefficients of the diagonal state probability generating function are shown to be related to ‘manila folder configurations in a filing cabinet’, and we make this connection precise. This allows us to use known results for the enumeration of such manila folder configurations to give a closed form expression for the diagonal state return probabilities.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102996"},"PeriodicalIF":1.3,"publicationDate":"2025-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145474010","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 : 2025-10-31DOI: 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":"2025-10-31","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 : 2025-10-21DOI: 10.1016/j.aam.2025.102984
Kristina Ago, Bojan Bašić
The so-called MP-ratio is a kind of measure of how “packed with palindromes” a given word is. The lower bound on the MP-ratio for the set of all n-ary words is (trivially) 1, while the best possible upper bound is an open problem in the general case. It is solved for (where the optimal upper bound is 4) and for (where the optimal upper bound is 6). Also, it is known that in the n-ary case the optimal bound is between 2n and the order of growth . In this article we solve this problem for quaternary words, for which we show that the best possible upper bound on the MP-ratio equals 8. We believe that this is the last case in which the result is 2n, that is, we believe that for there are words whose MP-ratio is strictly larger than 2n.
{"title":"The optimal upper bound on the MP-ratio for quaternary words","authors":"Kristina Ago, Bojan Bašić","doi":"10.1016/j.aam.2025.102984","DOIUrl":"10.1016/j.aam.2025.102984","url":null,"abstract":"<div><div>The so-called <em>MP-ratio</em> is a kind of measure of how “packed with palindromes” a given word is. The lower bound on the MP-ratio for the set of all <em>n</em>-ary words is (trivially) 1, while the best possible upper bound is an open problem in the general case. It is solved for <span><math><mi>n</mi><mo>=</mo><mn>2</mn></math></span> (where the optimal upper bound is 4) and for <span><math><mi>n</mi><mo>=</mo><mn>3</mn></math></span> (where the optimal upper bound is 6). Also, it is known that in the <em>n</em>-ary case the optimal bound is between 2<em>n</em> and the order of growth <span><math><mi>n</mi><msup><mrow><mn>2</mn></mrow><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></math></span>. In this article we solve this problem for quaternary words, for which we show that the best possible upper bound on the MP-ratio equals 8. We believe that this is the last case in which the result is 2<em>n</em>, that is, we believe that for <span><math><mi>n</mi><mo>⩾</mo><mn>5</mn></math></span> there are words whose MP-ratio is strictly larger than 2<em>n</em>.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102984"},"PeriodicalIF":1.3,"publicationDate":"2025-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145333347","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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