{"title":"Matroid Horn functions","authors":"Kristóf Bérczi , Endre Boros , Kazuhisa Makino","doi":"10.1016/j.jcta.2023.105838","DOIUrl":"https://doi.org/10.1016/j.jcta.2023.105838","url":null,"abstract":"<div><p>Hypergraph Horn functions were introduced as a subclass of Horn functions that can be represented by a collection of circular implication rules. These functions possess distinguished structural and computational properties. In particular, their characterizations in terms of implicate-duality and the closure operator provide extensions of matroid duality and the Mac Lane<!--> <!-->–<!--> <!-->Steinitz exchange property of matroid closure, respectively.</p><p>In the present paper, we introduce a subclass of hypergraph Horn functions that we call <em>matroid Horn</em> functions. We provide multiple characterizations of matroid Horn functions in terms of their canonical and complete CNF representations. We also study the Boolean minimization problem for this class, where the goal is to find a minimum size representation of a matroid Horn function given by a CNF representation. While there are various ways to measure the size of a CNF, we focus on the <em>number of circuits</em> and <em>circuit clauses</em>. We determine the size of an optimal representation for binary matroids, and give lower and upper bounds in the uniform case. For uniform matroids, we show a strong connection between our problem and Turán systems that might be of independent combinatorial interest.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"203 ","pages":"Article 105838"},"PeriodicalIF":1.1,"publicationDate":"2023-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0097316523001061/pdfft?md5=55c70db92d34f783b8e0189c2f8d7950&pid=1-s2.0-S0097316523001061-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138404270","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}
{"title":"Some refinements of Stanley's shuffle theorem","authors":"Kathy Q. Ji, Dax T.X. Zhang","doi":"10.1016/j.jcta.2023.105830","DOIUrl":"10.1016/j.jcta.2023.105830","url":null,"abstract":"<div><p>We give a combinatorial proof of Stanley's shuffle theorem by using the insertion lemma of Haglund, Loehr and Remmel. Based on this combinatorial construction, we establish several refinements of Stanley's shuffle theorem.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"203 ","pages":"Article 105830"},"PeriodicalIF":1.1,"publicationDate":"2023-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138289376","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}
{"title":"The Clebsch–Gordan coefficients of U(sl2) and the Terwilliger algebras of Johnson graphs","authors":"Hau-Wen Huang","doi":"10.1016/j.jcta.2023.105833","DOIUrl":"10.1016/j.jcta.2023.105833","url":null,"abstract":"<div><p><span>The universal enveloping algebra </span><span><math><mi>U</mi><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> of <span><math><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span><span> is a unital associative algebra over </span><span><math><mi>C</mi></math></span> generated by <span><math><mi>E</mi><mo>,</mo><mi>F</mi><mo>,</mo><mi>H</mi></math></span> subject to the relations<span><span><span><math><mrow><mo>[</mo><mi>H</mi><mo>,</mo><mi>E</mi><mo>]</mo><mo>=</mo><mn>2</mn><mi>E</mi><mo>,</mo><mspace></mspace><mo>[</mo><mi>H</mi><mo>,</mo><mi>F</mi><mo>]</mo><mo>=</mo><mo>−</mo><mn>2</mn><mi>F</mi><mo>,</mo><mspace></mspace><mo>[</mo><mi>E</mi><mo>,</mo><mi>F</mi><mo>]</mo><mo>=</mo><mi>H</mi><mo>.</mo></mrow></math></span></span></span> The element<span><span><span><math><mi>Λ</mi><mo>=</mo><mi>E</mi><mi>F</mi><mo>+</mo><mi>F</mi><mi>E</mi><mo>+</mo><mfrac><mrow><msup><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>2</mn></mrow></mfrac></math></span></span></span> is called the Casimir element of <span><math><mi>U</mi><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>. Let <span><math><mi>Δ</mi><mo>:</mo><mi>U</mi><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>→</mo><mi>U</mi><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>⊗</mo><mi>U</mi><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> denote the comultiplication of <span><math><mi>U</mi><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>. The universal Hahn algebra <span><math><mi>H</mi></math></span> is a unital associative algebra over <span><math><mi>C</mi></math></span> generated by <span><math><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi></math></span> and the relations assert that <span><math><mo>[</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>]</mo><mo>=</mo><mi>C</mi></math></span> and each of<span><span><span><math><mrow><mo>[</mo><mi>C</mi><mo>,</mo><mi>A</mi><mo>]</mo><mo>+</mo><mn>2</mn><msup><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>B</mi><mo>,</mo><mspace></mspace><mo>[</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo>]</mo><mo>+</mo><mn>4</mn><mi>B</mi><mi>A</mi><mo>+</mo><mn>2</mn><mi>C</mi></mrow></math></span></span></span> is central in <span><math><mi>H</mi></math></span>. Inspired by the Clebsch–Gordan coefficients of <span><math><mi>U</mi><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span><span>, we discover an algebra homomorphism </span><span><math><mo>♮</mo><mo>:</mo><mi>H</mi><mo>→</mo><mi>U</mi><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>⊗</mo><mi>U</mi><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> that maps<span><span><","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"203 ","pages":"Article 105833"},"PeriodicalIF":1.1,"publicationDate":"2023-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138289377","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}
{"title":"A modular approach to Andrews-Beck partition statistics","authors":"Renrong Mao","doi":"10.1016/j.jcta.2023.105832","DOIUrl":"10.1016/j.jcta.2023.105832","url":null,"abstract":"<div><p>Andrews recently provided a <em>q</em>-series proof of congruences for <span><math><mi>N</mi><mi>T</mi><mo>(</mo><mi>m</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span>, the total number of parts in the partitions of <em>n</em> with rank congruent to <em>m</em><span> modulo </span><em>k</em>. Motivated by Andrews' works, Chern obtain congruences for <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>ω</mi></mrow></msub><mo>(</mo><mi>m</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> which denotes the total number of ones in the partition of <em>n</em> with crank congruent to <em>m</em> modulo <em>k</em><span>. In this paper, we focus on the modular approach to these new partition statistics. Applying the theory of mock modular forms, we establish equalities and identities for </span><span><math><mi>N</mi><mi>T</mi><mo>(</mo><mi>m</mi><mo>,</mo><mn>7</mn><mo>,</mo><mi>n</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>ω</mi></mrow></msub><mo>(</mo><mi>m</mi><mo>,</mo><mn>7</mn><mo>,</mo><mi>n</mi><mo>)</mo></math></span>.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"203 ","pages":"Article 105832"},"PeriodicalIF":1.1,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"110423253","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}
Nicolas Crampé , Luc Vinet , Meri Zaimi , Xiaohong Zhang
{"title":"A bivariate Q-polynomial structure for the non-binary Johnson scheme","authors":"Nicolas Crampé , Luc Vinet , Meri Zaimi , Xiaohong Zhang","doi":"10.1016/j.jcta.2023.105829","DOIUrl":"10.1016/j.jcta.2023.105829","url":null,"abstract":"<div><p>The notion of multivariate <em>P</em>- and <em>Q</em><span>-polynomial association scheme has been introduced recently, generalizing the well-known univariate case<span>. Numerous examples of such association schemes have already been exhibited. In particular, it has been demonstrated that the non-binary Johnson scheme is a bivariate </span></span><em>P</em>-polynomial association scheme. We show here that it is also a bivariate <em>Q</em>-polynomial association scheme for some parameters. This provides, with the <em>P</em>-polynomial structure, the bispectral property (<em>i.e.</em><span> the recurrence and difference relations) of a family of bivariate orthogonal polynomials made out of univariate Krawtchouk and dual Hahn polynomials. The algebra based on the bispectral operators is also studied together with the subconstituent algebra of this association scheme.</span></p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"202 ","pages":"Article 105829"},"PeriodicalIF":1.1,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71509763","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}
{"title":"Non-expansive matrix number systems with bases similar to certain Jordan blocks","authors":"Joshua W. Caldwell , Kevin G. Hare , Tomáš Vávra","doi":"10.1016/j.jcta.2023.105828","DOIUrl":"https://doi.org/10.1016/j.jcta.2023.105828","url":null,"abstract":"<div><p>We study representations of integral vectors in a number system with a matrix base <em>M</em> and vector digits. We focus on the case when <em>M</em> is equal or similar to <span><math><msub><mrow><mi>J</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, the Jordan block with eigenvalue 1 and dimension <em>n</em>. If <span><math><mi>M</mi><mo>=</mo><msub><mrow><mi>J</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, we classify all digit sets of size two allowing representation for all of <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. For <span><math><mi>M</mi><mo>=</mo><msub><mrow><mi>J</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> with <span><math><mi>n</mi><mo>≥</mo><mn>3</mn></math></span>, we show that a digit set of size three suffice to represent all of <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. For bases <em>M</em> similar to <span><math><msub><mrow><mi>J</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span>, we construct a digit set of size <em>n</em> such that all of <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> is represented. The language of words representing the zero vector with <span><math><mi>M</mi><mo>=</mo><msub><mrow><mi>J</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and the digits <span><math><msup><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mo>±</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>T</mi></mrow></msup></math></span> is shown not to be context-free, but to be recognizable by a Turing machine with logarithmic memory.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"202 ","pages":"Article 105828"},"PeriodicalIF":1.1,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50187332","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}
{"title":"On some double Nahm sums of Zagier","authors":"Zhineng Cao , Hjalmar Rosengren , Liuquan Wang","doi":"10.1016/j.jcta.2023.105819","DOIUrl":"https://doi.org/10.1016/j.jcta.2023.105819","url":null,"abstract":"<div><p>Zagier provided eleven conjectural rank two examples for Nahm's problem. All of them have been proved in the literature except for the fifth example, and there is no <em>q</em>-series proof for the tenth example. We prove that the fifth and the tenth examples are in fact equivalent. Then we give a <em>q</em>-series proof for the fifth example, which confirms a recent conjecture of Wang. This also serves as the first <em>q</em><span>-series proof for the tenth example, whose explicit form was conjectured by Vlasenko and Zwegers in 2011 and whose modularity was proved by Cherednik and Feigin in 2013 via nilpotent double affine Hecke algebras.</span></p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"202 ","pages":"Article 105819"},"PeriodicalIF":1.1,"publicationDate":"2023-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50187330","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}
{"title":"Union-closed sets and Horn Boolean functions","authors":"Vadim Lozin , Viktor Zamaraev","doi":"10.1016/j.jcta.2023.105818","DOIUrl":"https://doi.org/10.1016/j.jcta.2023.105818","url":null,"abstract":"<div><p>A family <span><math><mi>F</mi></math></span> of sets is union-closed if the union of any two sets from <span><math><mi>F</mi></math></span> belongs to <span><math><mi>F</mi></math></span>. The union-closed sets conjecture states that if <span><math><mi>F</mi></math></span> is a finite union-closed family of finite sets, then there is an element that belongs to at least half of the sets in <span><math><mi>F</mi></math></span>. The conjecture has several equivalent formulations in terms of other combinatorial structures such as lattices and graphs. In its whole generality the conjecture remains wide open, but it was verified for various important classes of lattices, such as lower semimodular lattices, and graphs, such as chordal bipartite graphs. In the present paper we develop a Boolean approach to the conjecture and verify it for several classes of Boolean functions, such as submodular functions and double Horn functions.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"202 ","pages":"Article 105818"},"PeriodicalIF":1.1,"publicationDate":"2023-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50187331","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}
{"title":"Cycles of even-odd drop permutations and continued fractions of Genocchi numbers","authors":"Qiongqiong Pan , Jiang Zeng","doi":"10.1016/j.jcta.2023.105778","DOIUrl":"https://doi.org/10.1016/j.jcta.2023.105778","url":null,"abstract":"<div><p><span><span><span>Recently Lazar and Wachs proved two new permutation models, called D-permutations and E-permutations, for Genocchi and median Genocchi numbers. In a follow-up, Eu et al. studied the even-odd descent permutations, which are in </span>bijection with E-permutations. We generalize Eu et al.'s descent polynomials with eight </span>statistics and obtain an explicit J-fraction formula for their ordinary generaing function. The J-fraction permits us to confirm two conjectures of Lazar-Wachs about cycles of D and E permutations and obtain a </span><span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span>-analogue of Eu et al.'s gamma-formula. Moreover, the <span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span><span> gamma-coefficients have the same factorization flavor as the gamma-coefficients of Brändén's </span><span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span>-Eulerian polynomials.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"199 ","pages":"Article 105778"},"PeriodicalIF":1.1,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50185175","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}
Kangquan Li , Chunlei Li , Tor Helleseth , Longjiang Qu
{"title":"Further investigations on permutation based constructions of bent functions","authors":"Kangquan Li , Chunlei Li , Tor Helleseth , Longjiang Qu","doi":"10.1016/j.jcta.2023.105779","DOIUrl":"https://doi.org/10.1016/j.jcta.2023.105779","url":null,"abstract":"<div><p><span><span><span>Constructing bent functions by composing a Boolean function with a </span>permutation was introduced by Hou and Langevin in 1997. The approach appears simple but heavily depends on the construction of desirable permutations. In this paper, we further study this approach by investigating the exponential sums of certain </span>monomials and permutations. We propose several classes of bent functions from quadratic permutations and permutations with (generalized) Niho exponents, and also a class of bent functions from a generalization of the Maiorana-McFarland class. The relations among the proposed bent functions and the known families of bent function are studied. Numerical results show that our constructions include bent functions that are not contained in the completed Maiorana-McFarland class </span><span><math><msup><mrow><mi>M</mi></mrow><mrow><mi>#</mi></mrow></msup></math></span>, the class <span><math><msub><mrow><mi>PS</mi></mrow><mrow><mi>a</mi><mi>p</mi></mrow></msub></math></span> or the class <span><math><mi>H</mi></math></span>.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"199 ","pages":"Article 105779"},"PeriodicalIF":1.1,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50185176","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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