Pub Date : 2025-04-25DOI: 10.1016/j.jcta.2025.106059
Ishay Haviv
The storage capacity of a graph measures the maximum amount of information that can be stored across its vertices, such that the information at any vertex can be recovered from the information stored at its neighborhood. The study of this graph quantity is motivated by applications in distributed storage and by its intimate relations to the index coding problem from the area of network information theory. In the latter, one wishes to minimize the amount of information that has to be transmitted to a collection of receivers, in a way that enables each of them to discover its required data using some prior side information.
In this paper, we initiate the study of the and problems from the perspective of parameterized complexity. We prove that the problem parameterized by the solution size admits a kernelization algorithm producing kernels of linear size. We also provide such a result for the problem, in the linear and non-linear settings, where it is parameterized by the dual value of the solution, i.e., the length of the transmission that can be saved using the side information. A key ingredient in the proofs is the crown decomposition technique due to Chor, Fellows, and Juedes [14], [11]. As an application, we significantly extend an algorithmic result of Dau, Skachek, and Chee [13].
{"title":"Kernels for storage capacity and dual index coding","authors":"Ishay Haviv","doi":"10.1016/j.jcta.2025.106059","DOIUrl":"10.1016/j.jcta.2025.106059","url":null,"abstract":"<div><div>The storage capacity of a graph measures the maximum amount of information that can be stored across its vertices, such that the information at any vertex can be recovered from the information stored at its neighborhood. The study of this graph quantity is motivated by applications in distributed storage and by its intimate relations to the index coding problem from the area of network information theory. In the latter, one wishes to minimize the amount of information that has to be transmitted to a collection of receivers, in a way that enables each of them to discover its required data using some prior side information.</div><div>In this paper, we initiate the study of the <figure><img></figure> and <figure><img></figure> problems from the perspective of parameterized complexity. We prove that the <figure><img></figure> problem parameterized by the solution size admits a kernelization algorithm producing kernels of linear size. We also provide such a result for the <figure><img></figure> problem, in the linear and non-linear settings, where it is parameterized by the dual value of the solution, i.e., the length of the transmission that can be saved using the side information. A key ingredient in the proofs is the crown decomposition technique due to Chor, Fellows, and Juedes <span><span>[14]</span></span>, <span><span>[11]</span></span>. As an application, we significantly extend an algorithmic result of Dau, Skachek, and Chee <span><span>[13]</span></span>.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"216 ","pages":"Article 106059"},"PeriodicalIF":0.9,"publicationDate":"2025-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143868647","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-04-16DOI: 10.1016/j.jcta.2025.106051
Yuefeng Yang , Akihiro Munemasa , Kaishun Wang , Wenying Zhu
We classify certain non-symmetric commutative association schemes. As an application, we determine all the weakly distance-regular circulants of one type of arcs by using Schur rings. We also give the classification of primitive weakly distance-regular circulants.
{"title":"Weakly distance-regular circulants, I","authors":"Yuefeng Yang , Akihiro Munemasa , Kaishun Wang , Wenying Zhu","doi":"10.1016/j.jcta.2025.106051","DOIUrl":"10.1016/j.jcta.2025.106051","url":null,"abstract":"<div><div>We classify certain non-symmetric commutative association schemes. As an application, we determine all the weakly distance-regular circulants of one type of arcs by using Schur rings. We also give the classification of primitive weakly distance-regular circulants.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"216 ","pages":"Article 106051"},"PeriodicalIF":0.9,"publicationDate":"2025-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143834212","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-04-03DOI: 10.1016/j.jcta.2025.106047
Dean Crnković , Maarten De Boeck , Francesco Pavese , Andrea Švob
Divisible design graphs were introduced in 2011 by Haemers, Kharaghani and Meulenberg. In this paper, we introduce the notion of q-analogs of divisible design graphs and show that all q-analogs of divisible design graphs come from spreads, and are actually q-analogs of strongly regular graphs.
Deza graphs were introduced by Erickson, Fernando, Haemers, Hardy and Hemmeter in 1999. In this paper, we introduce q-analogs of Deza graphs. Further, we determine possible parameters, give examples of q-analogs of Deza graphs and characterize all non-strongly regular q-analogs of Deza graphs with the smallest parameters.
{"title":"q-Analogs of divisible design graphs and Deza graphs","authors":"Dean Crnković , Maarten De Boeck , Francesco Pavese , Andrea Švob","doi":"10.1016/j.jcta.2025.106047","DOIUrl":"10.1016/j.jcta.2025.106047","url":null,"abstract":"<div><div>Divisible design graphs were introduced in 2011 by Haemers, Kharaghani and Meulenberg. In this paper, we introduce the notion of <em>q</em>-analogs of divisible design graphs and show that all <em>q</em>-analogs of divisible design graphs come from spreads, and are actually <em>q</em>-analogs of strongly regular graphs.</div><div>Deza graphs were introduced by Erickson, Fernando, Haemers, Hardy and Hemmeter in 1999. In this paper, we introduce <em>q</em>-analogs of Deza graphs. Further, we determine possible parameters, give examples of <em>q</em>-analogs of Deza graphs and characterize all non-strongly regular <em>q</em>-analogs of Deza graphs with the smallest parameters.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"215 ","pages":"Article 106047"},"PeriodicalIF":0.9,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143761107","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-04-03DOI: 10.1016/j.jcta.2025.106046
Zhiguo Ding , Michael E. Zieve
For each odd prime power q, we describe a class of rational functions with the following unusual property: for every odd j, the function induced by on is 2-to-1. We also show that, among all known rational functions which are 2-to-1 on for infinitely many j, our new functions are the only ones which cannot be written as compositions of rational functions of degree at most four, monomials, Dickson polynomials, and additive (linearized) polynomials.
{"title":"Exceptional 2-to-1 rational functions","authors":"Zhiguo Ding , Michael E. Zieve","doi":"10.1016/j.jcta.2025.106046","DOIUrl":"10.1016/j.jcta.2025.106046","url":null,"abstract":"<div><div>For each odd prime power <em>q</em>, we describe a class of rational functions <span><math><mi>f</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> with the following unusual property: for every odd <em>j</em>, the function induced by <span><math><mi>f</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> on <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>j</mi></mrow></msup></mrow></msub><mo>∪</mo><mo>{</mo><mo>∞</mo><mo>}</mo></math></span> is 2-to-1. We also show that, among all known rational functions <span><math><mi>f</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> which are 2-to-1 on <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>j</mi></mrow></msup></mrow></msub><mo>∪</mo><mo>{</mo><mo>∞</mo><mo>}</mo></math></span> for infinitely many <em>j</em>, our new functions are the only ones which cannot be written as compositions of rational functions of degree at most four, monomials, Dickson polynomials, and additive (linearized) polynomials.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"215 ","pages":"Article 106046"},"PeriodicalIF":0.9,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143761106","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-04-03DOI: 10.1016/j.jcta.2025.106049
Zhicong Lin , Jing Liu , Sherry H.F. Yan
Motivated by Kitaev and Zhang's recent work on non-overlapping ascents in stack-sortable permutations and Dumont's permutation interpretation of the Jacobi elliptic functions, we investigate some parity statistics on restricted permutations. Some new related bijections are constructed and two refinements of the generating function for descents over 321-avoiding permutations due to Barnabei, Bonetti and Silimbanian are obtained. In particular, an open problem of Kitaev and Zhang about non-overlapping ascents on 321-avoiding permutations is solved and several combinatorial interpretations for the Catalan–Schett polynomials are found. The stack-sortable permutations are at the heart of our approaches.
{"title":"Parity statistics on restricted permutations and the Catalan–Schett polynomials","authors":"Zhicong Lin , Jing Liu , Sherry H.F. Yan","doi":"10.1016/j.jcta.2025.106049","DOIUrl":"10.1016/j.jcta.2025.106049","url":null,"abstract":"<div><div>Motivated by Kitaev and Zhang's recent work on non-overlapping ascents in stack-sortable permutations and Dumont's permutation interpretation of the Jacobi elliptic functions, we investigate some parity statistics on restricted permutations. Some new related bijections are constructed and two refinements of the generating function for descents over 321-avoiding permutations due to Barnabei, Bonetti and Silimbanian are obtained. In particular, an open problem of Kitaev and Zhang about non-overlapping ascents on 321-avoiding permutations is solved and several combinatorial interpretations for the Catalan–Schett polynomials are found. The stack-sortable permutations are at the heart of our approaches.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"215 ","pages":"Article 106049"},"PeriodicalIF":0.9,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143761108","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-04-03DOI: 10.1016/j.jcta.2025.106048
Shane Chern , Lin Jiu , Italo Simonelli
Given a positive integer n, consider a permutation of n objects chosen uniformly at random. In this permutation, we collect maximal subsequences consisting of consecutive numbers arranged in ascending order called blocks. Each block is then merged, and after all merges, the elements of this new set are relabeled from 1 to the current number of elements. We continue to permute and merge this new set uniformly at random until only one object is left. In this paper, we investigate the distribution of , the number of permutations needed for this process to end. In particular, we find explicit asymptotic expressions for the mean value , the variance , and higher central moments, and show that satisfies a central limit theorem.
{"title":"A central limit theorem for a card shuffling problem","authors":"Shane Chern , Lin Jiu , Italo Simonelli","doi":"10.1016/j.jcta.2025.106048","DOIUrl":"10.1016/j.jcta.2025.106048","url":null,"abstract":"<div><div>Given a positive integer <em>n</em>, consider a permutation of <em>n</em> objects chosen uniformly at random. In this permutation, we collect maximal subsequences consisting of consecutive numbers arranged in ascending order called blocks. Each block is then merged, and after all merges, the elements of this new set are relabeled from 1 to the current number of elements. We continue to permute and merge this new set uniformly at random until only one object is left. In this paper, we investigate the distribution of <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, the number of permutations needed for this process to end. In particular, we find explicit asymptotic expressions for the mean value <span><math><mi>E</mi><mo>[</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo></math></span>, the variance <span><math><mrow><mi>Var</mi></mrow><mo>[</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo></math></span>, and higher central moments, and show that <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> satisfies a central limit theorem.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"214 ","pages":"Article 106048"},"PeriodicalIF":0.9,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143759116","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let Γ be a Cayley graph on a finite group G, and let be the normalizer of (the right regular representation of G) in the full automorphism group of Γ. We say that Γ is a normal Cayley graph on G if , and that Γ is a normal edge-transitive Cayley graph on G if acts transitively on the edge set of Γ. In 1999, Praeger proved that every connected normal edge-transitive Cayley graph on a finite non-abelian simple group of valency 3 is normal. As an extension of this, in this paper, we prove that every connected normal edge-transitive Cayley graph on a finite non-abelian simple group of valency p is normal for each prime p. This, however, is not true for composite valency. We give a method to construct connected normal edge-transitive but non-normal Cayley graphs of certain groups, and using this, we prove that if G is either for an odd prime , or for , then there exists a connected normal edge-transitive but non-normal 8-valent Cayley graph of G.
{"title":"Normal edge-transitive Cayley graphs on non-abelian simple groups","authors":"Xing Zhang, Yan-Quan Feng, Fu-Gang Yin, Jin-Xin Zhou","doi":"10.1016/j.jcta.2025.106050","DOIUrl":"10.1016/j.jcta.2025.106050","url":null,"abstract":"<div><div>Let <em>Γ</em> be a Cayley graph on a finite group <em>G</em>, and let <span><math><msub><mrow><mi>N</mi></mrow><mrow><mrow><mi>Aut</mi></mrow><mo>(</mo><mi>Γ</mi><mo>)</mo></mrow></msub><mo>(</mo><mi>R</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo></math></span> be the normalizer of <span><math><mi>R</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> (the right regular representation of <em>G</em>) in the full automorphism group <span><math><mrow><mi>Aut</mi></mrow><mo>(</mo><mi>Γ</mi><mo>)</mo></math></span> of <em>Γ</em>. We say that <em>Γ</em> is a normal Cayley graph on <em>G</em> if <span><math><msub><mrow><mi>N</mi></mrow><mrow><mrow><mi>Aut</mi></mrow><mo>(</mo><mi>Γ</mi><mo>)</mo></mrow></msub><mo>(</mo><mi>R</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo><mo>=</mo><mrow><mi>Aut</mi></mrow><mo>(</mo><mi>Γ</mi><mo>)</mo></math></span>, and that <em>Γ</em> is a normal edge-transitive Cayley graph on <em>G</em> if <span><math><msub><mrow><mi>N</mi></mrow><mrow><mrow><mi>Aut</mi></mrow><mo>(</mo><mi>Γ</mi><mo>)</mo></mrow></msub><mo>(</mo><mi>R</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo></math></span> acts transitively on the edge set of <em>Γ</em>. In 1999, Praeger proved that every connected normal edge-transitive Cayley graph on a finite non-abelian simple group of valency 3 is normal. As an extension of this, in this paper, we prove that every connected normal edge-transitive Cayley graph on a finite non-abelian simple group of valency <em>p</em> is normal for each prime <em>p</em>. This, however, is not true for composite valency. We give a method to construct connected normal edge-transitive but non-normal Cayley graphs of certain groups, and using this, we prove that if <em>G</em> is either <span><math><msub><mrow><mi>PSL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math></span> for an odd prime <span><math><mi>q</mi><mo>≥</mo><mn>5</mn></math></span>, or <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> for <span><math><mi>n</mi><mo>≥</mo><mn>5</mn></math></span>, then there exists a connected normal edge-transitive but non-normal 8-valent Cayley graph of <em>G</em>.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"215 ","pages":"Article 106050"},"PeriodicalIF":0.9,"publicationDate":"2025-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143739311","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-03-18DOI: 10.1016/j.jcta.2025.106035
Heng Huat Chan , Song Heng Chan
In this article, we discuss finite versions of Euler's pentagonal number identity, the Rogers-Ramanujan identities and present new proofs of the finite versions of the Andrews-Gordon identity and the Bressoud identity. We also investigate the finite version of Garvan's generalizations of Dyson's rank and discover a new one-variable extension of the Andrews-Gordon identity.
{"title":"Finite versions of the Andrews–Gordon identity and Bressoud's identity","authors":"Heng Huat Chan , Song Heng Chan","doi":"10.1016/j.jcta.2025.106035","DOIUrl":"10.1016/j.jcta.2025.106035","url":null,"abstract":"<div><div>In this article, we discuss finite versions of Euler's pentagonal number identity, the Rogers-Ramanujan identities and present new proofs of the finite versions of the Andrews-Gordon identity and the Bressoud identity. We also investigate the finite version of Garvan's generalizations of Dyson's rank and discover a new one-variable extension of the Andrews-Gordon identity.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"215 ","pages":"Article 106035"},"PeriodicalIF":0.9,"publicationDate":"2025-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143643721","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-03-13DOI: 10.1016/j.jcta.2025.106045
Edwin R. van Dam , Jack H. Koolen , Yanzhen Xiong
An association scheme is called amorphic if every possible fusion of relations gives rise to a fusion scheme. We call a pair of relations fusing if fusing that pair gives rise to a fusion scheme. We define the fusing-relations graph on the set of relations, where a pair forms an edge if it fuses. We show that if the fusing-relations graph is connected but not a path, then the association scheme is amorphic. As a side result, we show that if an association scheme has at most one relation that is neither strongly regular of Latin square type nor strongly regular of negative Latin square type, then it is amorphic.
{"title":"Characterizations of amorphic schemes and fusions of pairs","authors":"Edwin R. van Dam , Jack H. Koolen , Yanzhen Xiong","doi":"10.1016/j.jcta.2025.106045","DOIUrl":"10.1016/j.jcta.2025.106045","url":null,"abstract":"<div><div>An association scheme is called amorphic if every possible fusion of relations gives rise to a fusion scheme. We call a pair of relations fusing if fusing that pair gives rise to a fusion scheme. We define the fusing-relations graph on the set of relations, where a pair forms an edge if it fuses. We show that if the fusing-relations graph is connected but not a path, then the association scheme is amorphic. As a side result, we show that if an association scheme has at most one relation that is neither strongly regular of Latin square type nor strongly regular of negative Latin square type, then it is amorphic.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"215 ","pages":"Article 106045"},"PeriodicalIF":0.9,"publicationDate":"2025-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143611456","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-28DOI: 10.1016/j.jcta.2025.106034
Tian Han , Sergey Kitaev , Philip B. Zhang
In this paper, we find distributions of the left-to-right maxima, right-to-left maxima, left-to-right minima and right-to-left-minima statistics on up-down and down-up permutations of even and odd lengths. We recover and generalize a result by Carlitz and Scoville, obtained in 1975, stating that the distribution of left-to-right maxima on down-up permutations of even length is given by . We also derive the joint distribution of the maxima (resp., minima) statistics, extending the scope of the respective results of Carlitz and Scoville, who obtain them in terms of certain systems of PDEs and recurrence relations. To accomplish this, we generalize a result of Kitaev and Remmel by deriving joint distributions involving non-maxima (resp., non-minima) statistics. Consequently, we refine classic enumeration results of André by introducing new q-analogues and -analogues for the number of alternating permutations.
Additionally, we verify Callan's conjecture (2012) that up-down permutations of even length fixed by reverse and complement are counted by the Springer numbers, thereby offering another combinatorial interpretation of these numbers. Furthermore, we propose two q-analogues and a -analogue of the Springer numbers. Lastly, we enumerate alternating permutations that avoid certain flat partially ordered patterns.
{"title":"Distribution of maxima and minima statistics on alternating permutations, Springer numbers, and avoidance of flat POPs","authors":"Tian Han , Sergey Kitaev , Philip B. Zhang","doi":"10.1016/j.jcta.2025.106034","DOIUrl":"10.1016/j.jcta.2025.106034","url":null,"abstract":"<div><div>In this paper, we find distributions of the left-to-right maxima, right-to-left maxima, left-to-right minima and right-to-left-minima statistics on up-down and down-up permutations of even and odd lengths. We recover and generalize a result by Carlitz and Scoville, obtained in 1975, stating that the distribution of left-to-right maxima on down-up permutations of even length is given by <span><math><msup><mrow><mo>(</mo><mi>sec</mi><mo></mo><mo>(</mo><mi>t</mi><mo>)</mo><mo>)</mo></mrow><mrow><mi>q</mi></mrow></msup></math></span>. We also derive the joint distribution of the maxima (resp., minima) statistics, extending the scope of the respective results of Carlitz and Scoville, who obtain them in terms of certain systems of PDEs and recurrence relations. To accomplish this, we generalize a result of Kitaev and Remmel by deriving joint distributions involving non-maxima (resp., non-minima) statistics. Consequently, we refine classic enumeration results of André by introducing new <em>q</em>-analogues and <span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span>-analogues for the number of alternating permutations.</div><div>Additionally, we verify Callan's conjecture (2012) that up-down permutations of even length fixed by reverse and complement are counted by the Springer numbers, thereby offering another combinatorial interpretation of these numbers. Furthermore, we propose two <em>q</em>-analogues and a <span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span>-analogue of the Springer numbers. Lastly, we enumerate alternating permutations that avoid certain flat partially ordered patterns.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"213 ","pages":"Article 106034"},"PeriodicalIF":0.9,"publicationDate":"2025-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143511458","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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