{"title":"Some conjectures of Ballantine and Merca on truncated sums and the minimal excludant in congruences classes","authors":"Olivia X.M. Yao","doi":"10.1016/j.jcta.2024.105967","DOIUrl":"10.1016/j.jcta.2024.105967","url":null,"abstract":"<div><div>In 2012, Andrews and Merca proved a truncated theorem on Euler's pentagonal number theorem. Since then, a number of results on truncated theta series have been proved. In this paper, we find the connections between truncated sums of certain partition functions and the minimal excludant statistic which has been found to exhibit connections with a handful of objects such as Dyson's crank. We present a uniform method to confirm five conjectures on truncated sums of certain partition functions given by Ballantine and Merca. In particular, we provide partition-theoretic interpretations for some truncated sums by using the minimal excludant in congruences classes.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"210 ","pages":"Article 105967"},"PeriodicalIF":0.9,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142586416","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":"Reconstruction of hypermatrices from subhypermatrices","authors":"Wenjie Zhong , Xiande Zhang","doi":"10.1016/j.jcta.2024.105966","DOIUrl":"10.1016/j.jcta.2024.105966","url":null,"abstract":"<div><div>For a given <em>n</em>, what is the smallest number <em>k</em> such that every sequence of length <em>n</em> is determined by the multiset of all its <em>k</em>-subsequences? This is called the <em>k</em>-deck problem for sequence reconstruction, and has been generalized to the two-dimensional case – reconstruction of <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span>-matrices from submatrices. Previous works show that the smallest <em>k</em> is at most <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo>)</mo></math></span> for sequences and at most <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mfrac><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup><mo>)</mo></math></span> for matrices. We study this <em>k</em>-deck problem for general dimension <em>d</em> and prove that, the smallest <em>k</em> is at most <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mfrac><mrow><mi>d</mi></mrow><mrow><mi>d</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow></msup><mo>)</mo></math></span> for reconstructing any <em>d</em> dimensional hypermatrix of order <em>n</em> from the multiset of all its subhypermatrices of order <em>k</em>.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"209 ","pages":"Article 105966"},"PeriodicalIF":0.9,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142530619","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":"Direct constructions of column-orthogonal strong orthogonal arrays","authors":"Jingjun Bao , Lijun Ji , Juanjuan Xu","doi":"10.1016/j.jcta.2024.105965","DOIUrl":"10.1016/j.jcta.2024.105965","url":null,"abstract":"<div><div>Strong orthogonal arrays have better space-filling properties than ordinary orthogonal arrays for computer experiments. Strong orthogonal arrays of strengths two plus, two star and three minus can improve the space-filling properties in low dimensions and column orthogonality plays a vital role in computer experiments. In this paper, we use difference matrices and generator matrices of linear codes to present several constructions of column-orthogonal strong orthogonal arrays of strengths two plus, two star, three minus and <em>t</em>. Our constructions can provide larger numbers of factors of column-orthogonal strong orthogonal arrays of strengths two plus, two star, three minus and <em>t</em> than those in the existing literature, enjoy flexible run sizes. These constructions are convenient, and the resulting designs are good choices for computer experiments.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"209 ","pages":"Article 105965"},"PeriodicalIF":0.9,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142530618","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}
Michael Fisher , Neil A. McKay , Rebecca Milley , Richard J. Nowakowski , Carlos P. Santos
{"title":"Indecomposable combinatorial games","authors":"Michael Fisher , Neil A. McKay , Rebecca Milley , Richard J. Nowakowski , Carlos P. Santos","doi":"10.1016/j.jcta.2024.105964","DOIUrl":"10.1016/j.jcta.2024.105964","url":null,"abstract":"<div><div>In Combinatorial Game Theory, short game forms are defined recursively over all the positions the two players are allowed to move to. A form is decomposable if it can be expressed as a disjunctive sum of two forms with smaller birthday. If there are no such summands, then the form is indecomposable. The main contribution of this document is the characterization of the indecomposable nimbers and the characterization of the indecomposable numbers. More precisely, a nimber is indecomposable if and only if its size is a power of two, and a number is indecomposable if and only if its absolute value is less than or equal to one.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"209 ","pages":"Article 105964"},"PeriodicalIF":0.9,"publicationDate":"2024-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142437901","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":"Point-line geometries related to binary equidistant codes","authors":"Mark Pankov , Krzysztof Petelczyc , Mariusz Żynel","doi":"10.1016/j.jcta.2024.105962","DOIUrl":"10.1016/j.jcta.2024.105962","url":null,"abstract":"<div><div>Point-line geometries whose singular subspaces correspond to binary equidistant codes are investigated. The main result is a description of automorphisms of these geometries. In some important cases, automorphisms induced by non-monomial linear automorphisms surprisingly arise.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"210 ","pages":"Article 105962"},"PeriodicalIF":0.9,"publicationDate":"2024-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142425591","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":"Neighborly partitions, hypergraphs and Gordon's identities","authors":"Pooneh Afsharijoo , Hussein Mourtada","doi":"10.1016/j.jcta.2024.105963","DOIUrl":"10.1016/j.jcta.2024.105963","url":null,"abstract":"<div><div>We prove a family of partition identities which is “dual” to the family of Andrews-Gordon's identities. These identities are inspired by a correspondence between a special type of partitions and “hypergraphs” and their proof uses combinatorial commutative algebra.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"210 ","pages":"Article 105963"},"PeriodicalIF":0.9,"publicationDate":"2024-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142425592","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 locally n × n grid graphs","authors":"Carmen Amarra , Wei Jin , Cheryl E. Praeger","doi":"10.1016/j.jcta.2024.105957","DOIUrl":"10.1016/j.jcta.2024.105957","url":null,"abstract":"<div><div>We investigate locally <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> grid graphs, that is, graphs in which the neighbourhood of any vertex is the Cartesian product of two complete graphs on <em>n</em> vertices. We consider the subclass of these graphs for which each pair of vertices at distance two is joined by sufficiently many paths of length 2. The number of such paths is known to be at most 2<em>n</em> by previous work of Blokhuis and Brouwer. We show that if each pair is joined by at least <span><math><mn>2</mn><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span> such paths then the diameter is at most 3 and we give a tight upper bound on the order of the graphs. We show that graphs meeting this upper bound are distance-regular antipodal covers of complete graphs. We exhibit an infinite family of such graphs which are locally <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> grid for odd prime powers <em>n</em>, and apply these results to locally <span><math><mn>5</mn><mo>×</mo><mn>5</mn></math></span> grid graphs to obtain a classification for the case where either all <em>μ</em>-graphs (induced subgraphs on the set of common neighbours of two vertices at distance two) have order at least 8 or all <em>μ</em>-graphs have order <em>c</em> for some constant <em>c</em>.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"209 ","pages":"Article 105957"},"PeriodicalIF":0.9,"publicationDate":"2024-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142322889","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 power monoids and their automorphisms","authors":"Salvatore Tringali, Weihao Yan","doi":"10.1016/j.jcta.2024.105961","DOIUrl":"10.1016/j.jcta.2024.105961","url":null,"abstract":"<div><div>Endowed with the binary operation of set addition, the family <span><math><msub><mrow><mi>P</mi></mrow><mrow><mrow><mi>fin</mi></mrow><mo>,</mo><mn>0</mn></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo></math></span> of all finite subsets of <span><math><mi>N</mi></math></span> containing 0 forms a monoid, with the singleton {0} as its neutral element.</div><div>We show that the only non-trivial automorphism of <span><math><msub><mrow><mi>P</mi></mrow><mrow><mrow><mi>fin</mi></mrow><mo>,</mo><mn>0</mn></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo></math></span> is the involution <span><math><mi>X</mi><mo>↦</mo><mi>max</mi><mo></mo><mi>X</mi><mo>−</mo><mi>X</mi></math></span>. The proof leverages ideas from additive number theory and proceeds through an unconventional induction on what we call the boxing dimension of a finite set of integers, that is, the smallest number of (discrete) intervals whose union is the set itself.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"209 ","pages":"Article 105961"},"PeriodicalIF":0.9,"publicationDate":"2024-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142319544","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 non-empty cross-t-intersecting families","authors":"Anshui Li , Huajun Zhang","doi":"10.1016/j.jcta.2024.105960","DOIUrl":"10.1016/j.jcta.2024.105960","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> be families of <em>k</em>-element subsets of a <em>n</em>-element set. We call them cross-<em>t</em>-intersecting if <span><math><mo>|</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∩</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>|</mo><mo>≥</mo><mi>t</mi></math></span> for any <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>∈</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span> with <span><math><mi>i</mi><mo>≠</mo><mi>j</mi></math></span>. In this paper we will prove that, for <span><math><mi>n</mi><mo>≥</mo><mn>2</mn><mi>k</mi><mo>−</mo><mi>t</mi><mo>+</mo><mn>1</mn></math></span>, if <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> are non-empty cross-<em>t</em>-intersecting families, then<span><span><span><math><munder><mo>∑</mo><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>m</mi></mrow></munder><mo>|</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo><mo>≤</mo><mi>max</mi><mo></mo><mo>{</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow><mo>−</mo><munder><mo>∑</mo><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>t</mi><mo>−</mo><mn>1</mn></mrow></munder><mrow><mo>(</mo><mtable><mtr><mtd><mi>k</mi></mtd></mtr><mtr><mtd><mi>i</mi></mtd></mtr></mtable><mo>)</mo></mrow><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></mtd></mtr><mtr><mtd><mrow><mi>k</mi><mo>−</mo><mi>i</mi></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mo>+</mo><mi>m</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>m</mi><mi>M</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>}</mo><mo>,</mo></math></span></span></span> where <span><math><mi>M</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span> is the size of the maximum <em>t</em>-intersecting family of <span><math><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></math></span>. Moreover, the extremal families attaining the upper bound are characterized.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"210 ","pages":"Article 105960"},"PeriodicalIF":0.9,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142316250","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":"Avoiding intersections of given size in finite affine spaces AG(n,2)","authors":"Benedek Kovács , Zoltán Lóránt Nagy","doi":"10.1016/j.jcta.2024.105959","DOIUrl":"10.1016/j.jcta.2024.105959","url":null,"abstract":"<div><div>We study the set of intersection sizes of a <em>k</em>-dimensional affine subspace and a point set of size <span><math><mi>m</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>]</mo></math></span> of the <em>n</em>-dimensional binary affine space <span><math><mrow><mi>AG</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mn>2</mn><mo>)</mo></math></span>. Following the theme of Erdős, Füredi, Rothschild and T. Sós, we partially determine which local densities in <em>k</em>-dimensional affine subspaces are unavoidable in all <em>m</em>-element point sets in the <em>n</em>-dimensional affine space.</div><div>We also show constructions of point sets for which the intersection sizes with <em>k</em>-dimensional affine subspaces take values from a set of a small size compared to <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>k</mi></mrow></msup></math></span>. These are built up from affine subspaces and so-called subspace evasive sets. Meanwhile, we improve the best known upper bounds on subspace evasive sets and apply results concerning the canonical signed-digit (CSD) representation of numbers.</div><div><em>Keywords</em>: unavoidable, affine subspaces, evasive sets, random methods, canonical signed-digit number system.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"209 ","pages":"Article 105959"},"PeriodicalIF":0.9,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0097316524000980/pdfft?md5=62687b67d599290d3f204041642a9a6a&pid=1-s2.0-S0097316524000980-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142314104","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}