Journal of Combinatorial Theory Series A最新文献

{"title":"Two-geodesic transitive graphs of order pn with n ≤ 3","authors":"Jun-Jie Huang,&nbsp;Yan-Quan Feng,&nbsp;Jin-Xin Zhou,&nbsp;Fu-Gang Yin","doi":"10.1016/j.jcta.2023.105814","DOIUrl":"https://doi.org/10.1016/j.jcta.2023.105814","url":null,"abstract":"<div><p>A vertex triple <span><math><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo>)</mo></math></span> of a graph is called a 2<em>-geodesic</em> if <em>v</em> is adjacent to both <em>u</em> and <em>w</em> and <em>u</em> is not adjacent to <em>w</em>. A graph is said to be 2<em>-geodesic transitive</em><span> if its automorphism group is transitive on the set of 2-geodesics. In this paper, a complete classification of 2-geodesic transitive graphs of order </span><span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> is given for each prime <em>p</em> and <span><math><mi>n</mi><mo>≤</mo><mn>3</mn></math></span><span>. It turns out that all such graphs consist of three small graphs: the complete bipartite graph </span><span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>4</mn><mo>,</mo><mn>4</mn></mrow></msub></math></span> of order 8, the Schläfli graph of order 27 and its complement, and fourteen infinite families: the cycles <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math></span> and <span><math><msub><mrow><mi>C</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></msub></math></span>, the complete graphs <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math></span> and <span><math><msub><mrow><mi>K</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></msub></math></span>, the complete multipartite graphs <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>p</mi><mo>[</mo><mi>p</mi><mo>]</mo></mrow></msub></math></span>, <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>p</mi><mo>[</mo><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>]</mo></mrow></msub></math></span> and <span><math><msub><mrow><mi>K</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>[</mo><mi>p</mi><mo>]</mo></mrow></msub></math></span>, the Hamming graph <span><math><mi>H</mi><mo>(</mo><mn>2</mn><mo>,</mo><mi>p</mi><mo>)</mo></math></span> and its complement, the Hamming graph <span><math><mi>H</mi><mo>(</mo><mn>3</mn><mo>,</mo><mi>p</mi><mo>)</mo></math></span><span>, and two infinite families of normal Cayley graphs on the extraspecial group of order </span><span><math><msup><mrow><mi>p</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> and exponent <em>p</em>.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50187327","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 bijection for length-5 patterns in permutations","authors":"Joanna N. Chen ,&nbsp;Zhicong Lin","doi":"10.1016/j.jcta.2023.105815","DOIUrl":"https://doi.org/10.1016/j.jcta.2023.105815","url":null,"abstract":"<div><p><span><span>A bijection<span> which preserves five classical set-valued permutation </span></span>statistics between </span><span><math><mo>(</mo><mn>31245</mn><mo>,</mo><mn>32145</mn><mo>,</mo><mn>31254</mn><mo>,</mo><mn>32154</mn><mo>)</mo></math></span>-avoiding permutations and <span><math><mo>(</mo><mn>31425</mn><mo>,</mo><mn>32415</mn><mo>,</mo><mn>31524</mn><mo>,</mo><mn>32514</mn><mo>)</mo></math></span>-avoiding permutations is constructed. Combining this bijection with two codings of permutations introduced respectively by Baril–Vajnovszki and Martinez–Savage, we prove an enumerative conjecture posed by Gao and Kitaev. Moreover, the generating function for the common counting sequence is proved to be algebraic.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50187329","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":"Paths of given length in tournaments","authors":"Sah, Ashwin, Sawhney, Mehtaab, Zhao, Yufei","doi":"10.5070/c63261981","DOIUrl":"https://doi.org/10.5070/c63261981","url":null,"abstract":"Author(s): Sah, Ashwin; Sawhney, Mehtaab; Zhao, Yufei | Abstract: We prove that every (n)-vertex tournament has at most (n left(frac{n-1}{2} right)^k) walks of length (k).Mathematics Subject Classifications: 05C38, 05D99Keywords: Paths, tournaments","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135488357","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":"Optimal schemes for combinatorial query problems with integer feedback","authors":"Anders Martinsson","doi":"10.5070/c63261997","DOIUrl":"https://doi.org/10.5070/c63261997","url":null,"abstract":"A query game is a pair of a set (Q) of queries and a set (mathcal{F}) of functions, or codewords (f:Qrightarrow mathbb{Z}.) We think of this as a two-player game. One player, Codemaker, picks a hidden codeword (fin mathcal{F}). The other player, Codebreaker, then tries to determine (f) by asking a sequence of queries (qin Q), after each of which Codemaker must respond with the value (f(q)). The goal of Codebreaker is to uniquely determine (f) using as few queries as possible. Two classical examples of such games are coin-weighing with a spring scale, and Mastermind, which are of interest both as recreational games and for their connection to information theory.In this paper, we will present a general framework for finding short solutions to query games. As applications, we give new self-contained proofs of the query complexity of variations of the coin-weighing problems, and prove new results that the deterministic query complexity of Mastermind with (n) positions and (k) colors is (Theta(n log k/ log n + k)) if only black-peg information is provided, and (Theta(n log k / log n + k/n)) if both black- and white-peg information is provided. In the deterministic setting, these are the first up to constant factor optimal solutions to Mastermind known for any (kgeq n^{1-o(1)}).Mathematics Subject Classifications: 91A46, 68Q11, 05B99Keywords: Combinatorial games, query complexity, Mastermind, coin-weighing","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134913100","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 Schwarzian octahedron recurrence (dSKP equation) I: explicit solutions","authors":"Niklas Affolter, Béatrice de Tilière, Paul Melotti","doi":"10.5070/c63261993","DOIUrl":"https://doi.org/10.5070/c63261993","url":null,"abstract":"We prove an explicit expression for the solutions of the discrete Schwarzian octahedron recurrence, also known as the discrete Schwarzian KP equation (dSKP), as the ratio of two partition functions. Each one counts weighted oriented dimer configurations of an associated bipartite graph, and is equal to the determinant of a Kasteleyn matrix. This is in the spirit of Speyer's result on the dKP equation, or octahedron recurrence [Spe07]. One consequence is that dSKP has zero algebraic entropy, meaning that the growth of the degrees of the polynomials involved is only polynomial. There are cancellations in the partition function, and we prove an alternative, cancellation free explicit expression involving complementary trees and forests. Using all of the above, we show several instances of the Devron property for dSKP, i.e., that certain singularities in initial data repeat after a finite number of steps. This has many applications for discrete geometric systems and is the subject of the companion paper [AdTM22]. We also prove limit shape results analogous to the arctic circle of the Aztec diamond. Finally, we discuss the combinatorics of all the other octahedral equations in the classification of Adler, Bobenko and Suris [ABS12].","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135488359","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":"Rectangular analogues of the square paths conjecture and the univariate Delta conjecture","authors":"Alessandro Iraci, Roberto Pagaria, Giovanni Paolini, Anna Vanden Wyngaerd","doi":"10.5070/c63261980","DOIUrl":"https://doi.org/10.5070/c63261980","url":null,"abstract":"In this paper, we extend the rectangular side of the shuffle conjecture by stating a rectangular analogue of the square paths conjecture. In addition, we describe a set of combinatorial objects and one statistic that are a first step towards a rectangular extension of (the rise version of) the Delta conjecture, and of (the rise version of) the Delta square conjecture, corresponding to the case (q=1) of an expected general statement. We also prove our new rectangular paths conjecture in the special case when the sides of the rectangle are coprime.Mathematics Subject Classifications: 05E05Keywords: Macdonald polynomials, symmetric functions","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134912783","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":"Mullineux involution and crystal isomorphisms","authors":"Nicolas Jacon","doi":"10.5070/c63261986","DOIUrl":"https://doi.org/10.5070/c63261986","url":null,"abstract":"We develop a new approach for the computation of the Mullineux involution for the symmetric group and its Hecke algebra using the notion of crystal isomorphism and the Iwahori-Matsumoto involution for the affine Hecke algebra of type (A). As a consequence, we obtain several new elementary combinatorial algorithms for its computation, one of which is equivalent to Xu algorithm (and thus Mullineux original algorithm). We thus obtain a simple interpretation of these algorithms and a new elementary proof that they indeed compute the Mullineux involution.Mathematics Subject Classifications: 20C08, 05E10Keywords: Symmetric group, Mullineux involution, crystal graph","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135487473","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":"Discrete dynamics in cluster integrable systems from geometric (R)-matrix transformations","authors":"Terrence George, Sanjay Ramassamy","doi":"10.5070/c63261990","DOIUrl":"https://doi.org/10.5070/c63261990","url":null,"abstract":"Cluster integrable systems are a broad class of integrable systems modelled on bipartite dimer models on the torus. Many discrete integrable dynamics arise by applying sequences of local transformations, which form the cluster modular group of the cluster integrable system. This cluster modular group was recently characterized by the first author and Inchiostro. There exist some discrete integrable dynamics that make use of non-local transformations associated with geometric (R)-matrices. In this article we characterize the generalized cluster modular group - which includes both local and non-local transformations - in terms of extended affine symmetric groups. We also describe the action of the generalized cluster modular group on the spectral data associated with cluster integrable systems.Mathematics Subject Classifications: 82B23, 13F60, 14H70, 20B35Keywords: Bipartite dimer model, cluster algebras, geometric (R)-matrices, discrete integrable systems, extended affine symmetric group","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135488360","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":"RSK tableaux and box-ball systems","authors":"Ben Drucker, Eli Garcia, Emily Gunawan, Aubrey Rumbolt, Rose Silver","doi":"10.5070/c63261978","DOIUrl":"https://doi.org/10.5070/c63261978","url":null,"abstract":"A box-ball system is a discrete dynamical system whose dynamics come from the balls jumping according to certain rules. A permutation on (n) objects gives a box-ball system state by assigning its one-line notation to (n) consecutive boxes. After a finite number of steps, a box-ball system will reach a steady state. From any steady state, we can construct a tableau called the soliton decomposition of the box-ball system. We prove that if the soliton decomposition of a permutation (w) is a standard tableau or if its shape coincides with the Robinson-Schensted (RS) partition of (w), then the soliton decomposition of (w) and the RS insertion tableau of (w) are equal. We also use row reading words, Knuth moves, RS recording tableaux, and a localized version of Greene's theorem (proven recently by Lewis, Lyu, Pylyavskyy, and Sen) to study various properties of a box-ball system.Mathematics Subject Classifications: 05A05, 05A17, 37B15Keywords: Permutations, box-ball systems, soliton cellular automata, Young tableaux, Robinson-Schensted-Knuth correspondence, Greene's theorem, Knuth equivalence","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134912788","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":"Generalized weights of codes over rings and invariants of monomial ideals","authors":"Elisa Gorla, Alberto Ravagnani","doi":"10.5070/c63261989","DOIUrl":"https://doi.org/10.5070/c63261989","url":null,"abstract":"We develop an algebraic theory of supports for (R)-linear codes of fixed length, where (R) is a finite commutative unitary ring. A support naturally induces a notion of generalized weights and allows one to associate a monomial ideal to a code. Our main result states that, under suitable assumptions, the generalized weights of a code can be obtained from the graded Betti numbers of its associated monomial ideal. In the case of (mathbb{F}_q)-linear codes endowed with the Hamming metric, the ideal coincides with the Stanley-Reisner ideal of the matroid associated to the code via its parity-check matrix. In this special setting, we recover the known result that the generalized weights of an (mathbb{F}_q)-linear code can be obtained from the graded Betti numbers of the ideal of the matroid associated to the code. We also study subcodes and codewords of minimal support in a code, proving that a large class of (R)-linear codes is generated by its codewords of minimal support.Mathematics Subject Classifications: 94B05, 13D02, 13F10Keywords: Linear codes, codes over rings, supports, generalized weights, monomial ideal of a code, graded Betti numbers, matroid","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134912401","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}