{"title":"On a conjecture concerning the r-Euler-Mahonian statistic on permutations","authors":"Kaimei Huang , Zhicong Lin , Sherry H.F. Yan","doi":"10.1016/j.jcta.2025.106008","DOIUrl":"10.1016/j.jcta.2025.106008","url":null,"abstract":"<div><div>A pair <span><math><mo>(</mo><mrow><mi>s</mi><msub><mrow><mi>t</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mo>,</mo><mrow><mi>s</mi><msub><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mo>)</mo></math></span> of permutation statistics is said to be <em>r</em>-Euler-Mahonian if <span><math><mo>(</mo><mrow><mi>s</mi><msub><mrow><mi>t</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mo>,</mo><mrow><mi>s</mi><msub><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mo>)</mo></math></span> and (rdes, rmaj) are equidistributed over the set <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> of all permutations of <span><math><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></math></span>, where rdes denotes the <em>r</em>-descent number and rmaj denotes the <em>r</em>-major index introduced by Rawlings. The main objective of this paper is to prove that <span><math><mo>(</mo><msub><mrow><mi>exc</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><msub><mrow><mi>den</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>)</mo></math></span> and (rdes, rmaj) are equidistributed over <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, thereby confirming a recent conjecture posed by Liu. When <span><math><mi>r</mi><mo>=</mo><mn>1</mn></math></span>, the result recovers the equidistribution of <span><math><mo>(</mo><mrow><mi>des</mi></mrow><mo>,</mo><mrow><mi>maj</mi></mrow><mo>)</mo></math></span> and <span><math><mo>(</mo><mrow><mi>exc</mi></mrow><mo>,</mo><mrow><mi>den</mi></mrow><mo>)</mo></math></span>, which was first conjectured by Denert and proved by Foata and Zeilberger.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"212 ","pages":"Article 106008"},"PeriodicalIF":0.9,"publicationDate":"2025-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143095725","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}
Jason P. Bell, Sean Monahan, Matthew Satriano, Karen Situ, Zheng Xie
{"title":"There are no good infinite families of toric codes","authors":"Jason P. Bell, Sean Monahan, Matthew Satriano, Karen Situ, Zheng Xie","doi":"10.1016/j.jcta.2025.106009","DOIUrl":"10.1016/j.jcta.2025.106009","url":null,"abstract":"<div><div>Soprunov and Soprunova posed a question on the existence of infinite families of toric codes that are “good” in a precise sense. We prove that such good families do not exist by proving a more general Szemerédi-type result: for all <span><math><mi>c</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span> and all positive integers <em>N</em>, subsets of density at least <em>c</em> in <span><math><msup><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>N</mi><mo>−</mo><mn>1</mn><mo>}</mo></mrow><mrow><mi>n</mi></mrow></msup></math></span> contain hypercubes of arbitrarily large dimension as <em>n</em> grows.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"213 ","pages":"Article 106009"},"PeriodicalIF":0.9,"publicationDate":"2025-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143103292","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":"Unique representations of integers by linear forms","authors":"Sándor Z. Kiss , Csaba Sándor","doi":"10.1016/j.jcta.2025.106007","DOIUrl":"10.1016/j.jcta.2025.106007","url":null,"abstract":"<div><div>Let <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> be an integer and let <em>A</em> be a set of nonnegative integers. For a <em>k</em>-tuple of positive integers <span><math><munder><mrow><mi>λ</mi></mrow><mo>_</mo></munder><mo>=</mo><mo>(</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span> with <span><math><mn>1</mn><mo>≤</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo><</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo><</mo><mo>⋯</mo><mo><</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>, we define the additive representation function <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>A</mi><mo>,</mo><munder><mrow><mi>λ</mi></mrow><mo>_</mo></munder></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mo>|</mo><mo>{</mo><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo><mo>∈</mo><msup><mrow><mi>A</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>:</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><mi>n</mi><mo>}</mo><mo>|</mo></math></span>. For <span><math><mi>k</mi><mo>=</mo><mn>2</mn></math></span>, Moser constructed a set <em>A</em> of nonnegative integers such that <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>A</mi><mo>,</mo><munder><mrow><mi>λ</mi></mrow><mo>_</mo></munder></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span> holds for every nonnegative integer <em>n</em>. In this paper we characterize all the <em>k</em>-tuples <span><math><munder><mrow><mi>λ</mi></mrow><mo>_</mo></munder></math></span> and the sets <em>A</em> of nonnegative integers with <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>A</mi><mo>,</mo><munder><mrow><mi>λ</mi></mrow><mo>_</mo></munder></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span> for every integer <span><math><mi>n</mi><mo>≥</mo><mn>0</mn></math></span>.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"213 ","pages":"Article 106007"},"PeriodicalIF":0.9,"publicationDate":"2025-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143099170","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":"Full weight spectrum one-orbit cyclic subspace codes","authors":"Chiara Castello, Olga Polverino, Ferdinando Zullo","doi":"10.1016/j.jcta.2024.106005","DOIUrl":"10.1016/j.jcta.2024.106005","url":null,"abstract":"<div><div>For a linear Hamming metric code of length <em>n</em> over a finite field, the number of distinct weights of its codewords is at most <em>n</em>. The codes achieving the equality in the above bound were called full weight spectrum codes. In this paper, we will focus on the analogous class of codes within the framework of cyclic subspace codes. Cyclic subspace codes have garnered significant attention, particularly for their applications in random network coding to correct errors and erasures. We investigate one-orbit cyclic subspace codes that are <em>full weight spectrum</em> in this context. Utilizing number-theoretical results and combinatorial arguments, we provide a complete classification of full weight spectrum one-orbit cyclic subspace codes.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"212 ","pages":"Article 106005"},"PeriodicalIF":0.9,"publicationDate":"2024-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143095723","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":"Contributions to Ma's conjecture concerning abelian difference sets with multiplier −1 (I)","authors":"Yasutsugu Fujita , Maohua Le","doi":"10.1016/j.jcta.2024.106004","DOIUrl":"10.1016/j.jcta.2024.106004","url":null,"abstract":"<div><div>Let <span><math><mi>N</mi></math></span>, <span><math><mi>P</mi></math></span> be the sets of all positive integers and odd primes, respectively. In 1991, when studying the existence of abelian difference sets with multiplier −1, S.-L. Ma <span><span>[14]</span></span> conjectured that the equation <span><math><mo>(</mo><mo>⁎</mo><mo>)</mo></math></span> <span><math><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mn>2</mn><mi>a</mi><mo>+</mo><mn>2</mn></mrow></msup><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>−</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>a</mi><mo>+</mo><mn>2</mn></mrow></msup><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi><mo>+</mo><mi>n</mi></mrow></msup><mo>+</mo><mn>1</mn></math></span>, <span><math><mi>p</mi><mo>∈</mo><mi>P</mi><mo>,</mo><mspace></mspace><mi>x</mi><mo>,</mo><mi>z</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>∈</mo><mi>N</mi></math></span> has only one solution <span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>a</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>)</mo><mo>=</mo><mo>(</mo><mn>5</mn><mo>,</mo><mn>49</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>. This is a far from solved problem that has been poorly known for so long. In this paper, using some elementary methods, we first prove that if <span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>a</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> is a solution of <span><math><mo>(</mo><mo>⁎</mo><mo>)</mo></math></span> with <span><math><mi>m</mi><mo>=</mo><mn>2</mn><mi>n</mi></math></span>, then there exist an odd positive integer <em>g</em> and a positive integer <em>t</em> which make <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>a</mi></mrow></msup><mo>=</mo><mo>(</mo><mi>g</mi><mo>+</mo><mn>1</mn><mo>)</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>−</mo><mo>(</mo><mi>g</mi><mo>−</mo><mn>1</mn><mo>)</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span> and <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>=</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span>, where <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>=</mo><mo>(</mo><msup><mrow><mi>α</mi></mrow><mrow><mi>r</mi></mrow></msup><mo>−</mo><msup><mrow><mover><mrow><mi>α</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mi>r</mi></mrow></msup><mo>)</mo><mo>/</mo><mo>(</mo><mi>α</mi><mo>−</mo><mover><mrow><mi>α</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>)</mo></math></span> for any integer <em>r</em>, <span><math><mi>α</mi><mo>=</mo><mn>2</mn><mi>g</mi><mo>+</mo><msqrt><mrow><mn>4</mn><msup><mrow><mi>g</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>1</mn></mrow></msqrt","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"212 ","pages":"Article 106004"},"PeriodicalIF":0.9,"publicationDate":"2024-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143100974","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":"Sidon sets, thin sets, and the nonlinearity of vectorial Boolean functions","authors":"Gábor P. Nagy","doi":"10.1016/j.jcta.2024.106001","DOIUrl":"10.1016/j.jcta.2024.106001","url":null,"abstract":"<div><div>The vectorial nonlinearity of a vector-valued function is its distance from the set of affine functions. In 2017, Liu, Mesnager, and Chen conjectured a general upper bound for the vectorial linearity. Recently, Carlet established a lower bound in terms of differential uniformity. In this paper, we improve Carlet's lower bound. Our approach is based on the fact that the level sets of a vectorial Boolean function are thin sets. In particular, level sets of APN functions are Sidon sets, hence the Liu-Mesnager-Chen conjecture predicts that in <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span>, there should be Sidon sets of size at least <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mo>/</mo><mn>2</mn></mrow></msup><mo>+</mo><mn>1</mn></math></span> for all <em>n</em>. This paper provides an overview of the known large Sidon sets in <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span>, and examines the completeness of the large Sidon sets derived from hyperbolas and ellipses of the finite affine plane.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"212 ","pages":"Article 106001"},"PeriodicalIF":0.9,"publicationDate":"2024-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142874013","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":"Diametric problem for permutations with the Ulam metric (optimal anticodes)","authors":"Pat Devlin, Leo Douhovnikoff","doi":"10.1016/j.jcta.2024.106002","DOIUrl":"10.1016/j.jcta.2024.106002","url":null,"abstract":"<div><div>We study the diametric problem (i.e., optimal anticodes) in the space of permutations under the Ulam distance. That is, let <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> denote the set of permutations on <em>n</em> symbols, and for each <span><math><mi>σ</mi><mo>,</mo><mi>τ</mi><mo>∈</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, define their Ulam distance as the number of distinct symbols that must be deleted from each until they are equal. We obtain a near-optimal upper bound on the size of the intersection of two balls in this space, and as a corollary, we prove that a set of diameter at most <em>k</em> has size at most <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>k</mi><mo>+</mo><mi>C</mi><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></msup><mi>n</mi><mo>!</mo><mo>/</mo><mo>(</mo><mi>n</mi><mo>−</mo><mi>k</mi><mo>)</mo><mo>!</mo></math></span>, compared to the best known construction of size <span><math><mi>n</mi><mo>!</mo><mo>/</mo><mo>(</mo><mi>n</mi><mo>−</mo><mi>k</mi><mo>)</mo><mo>!</mo></math></span>.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"212 ","pages":"Article 106002"},"PeriodicalIF":0.9,"publicationDate":"2024-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143095724","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":"Cayley extensions of maniplexes and polytopes","authors":"Gabe Cunningham , Elías Mochán , Antonio Montero","doi":"10.1016/j.jcta.2024.106000","DOIUrl":"10.1016/j.jcta.2024.106000","url":null,"abstract":"<div><div>A map on a surface whose automorphism group has a subgroup acting regularly on its vertices is called a Cayley map. Here we generalize that notion to maniplexes and polytopes. We define <span><math><mi>M</mi></math></span> to be a <em>Cayley extension</em> of <span><math><mi>K</mi></math></span> if the facets of <span><math><mi>M</mi></math></span> are isomorphic to <span><math><mi>K</mi></math></span> and if some subgroup of the automorphism group of <span><math><mi>M</mi></math></span> acts regularly on the facets of <span><math><mi>M</mi></math></span>. We show that many natural extensions in the literature on maniplexes and polytopes are in fact Cayley extensions. We also describe several universal Cayley extensions. Finally, we examine the automorphism group and symmetry type graph of Cayley extensions.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"212 ","pages":"Article 106000"},"PeriodicalIF":0.9,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143095348","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}
Björn Kriepke, Gohar M. Kyureghyan, Matthias Schymura
{"title":"On the size of integer programs with bounded non-vanishing subdeterminants","authors":"Björn Kriepke, Gohar M. Kyureghyan, Matthias Schymura","doi":"10.1016/j.jcta.2024.106003","DOIUrl":"10.1016/j.jcta.2024.106003","url":null,"abstract":"<div><div>Motivated by complexity questions in integer programming, this paper aims to contribute to the understanding of combinatorial properties of integer matrices of row rank <em>r</em> and with bounded subdeterminants. In particular, we study the column number question for integer matrices whose every <span><math><mi>r</mi><mo>×</mo><mi>r</mi></math></span> minor is non-zero and bounded by a fixed constant Δ in absolute value. Approaching the problem in two different ways, one that uses results from coding theory, and the other from the geometry of numbers, we obtain linear and asymptotically sublinear upper bounds on the maximal number of columns of such matrices, respectively. We complement these results by lower bound constructions, matching the linear upper bound for <span><math><mi>r</mi><mo>=</mo><mn>2</mn></math></span>, and a discussion of a computational approach to determine the maximal number of columns for small parameters Δ and <em>r</em>.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"212 ","pages":"Article 106003"},"PeriodicalIF":0.9,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143095349","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":"On joint short minimal zero-sum subsequences over finite abelian groups of rank two","authors":"Yushuang Fan , Qinghai Zhong","doi":"10.1016/j.jcta.2024.105984","DOIUrl":"10.1016/j.jcta.2024.105984","url":null,"abstract":"<div><div>Let <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mo>+</mo><mo>,</mo><mn>0</mn><mo>)</mo></math></span> be a finite abelian group and let <span><math><msup><mrow><mi>η</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the smallest integer <em>ℓ</em> such that every sequence over <span><math><mi>G</mi><mo>∖</mo><mo>{</mo><mn>0</mn><mo>}</mo></math></span> of length <em>ℓ</em> has two joint short minimal zero-sum subsequences. In 2013, Gao et al. obtained that <span><math><msup><mrow><mi>η</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>⊕</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>=</mo><mn>3</mn><mi>n</mi><mo>+</mo><mn>1</mn></math></span> for every <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span> and solved the corresponding inverse problem for groups <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>⊕</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>, where <em>p</em> is a prime. In this paper, we determine the precise value of <span><math><msup><mrow><mi>η</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo></math></span> for all finite abelian groups of rank 2 and resolve the corresponding inverse problem for groups <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>⊕</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, where <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span>, which confirms a conjecture of Gao, Geroldinger and Wang for all <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span> except <span><math><mi>n</mi><mo>=</mo><mn>4</mn></math></span>.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"212 ","pages":"Article 105984"},"PeriodicalIF":0.9,"publicationDate":"2024-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142790045","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}
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