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Birational geometry of generalized Hessenberg varieties and the generalized Shareshian-Wachs conjecture J. Comb. Theory A (IF 0.9) Pub Date : 2024-03-04 Young-Hoon Kiem, Donggun Lee
We introduce generalized Hessenberg varieties and establish basic facts. We show that the Tymoczko action of the symmetric group on the cohomology of Hessenberg varieties extends to generalized Hessenberg varieties and that natural morphisms among them preserve the action. By analyzing natural morphisms and birational maps among generalized Hessenberg varieties, we give an elementary proof of the Shareshian-Wachs
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The second largest eigenvalue of normal Cayley graphs on symmetric groups generated by cycles J. Comb. Theory A (IF 0.9) Pub Date : 2024-03-01 Yuxuan Li, Binzhou Xia, Sanming Zhou
We study the normal Cayley graphs on the symmetric group , where and is the set of all cycles in with length in . We prove that the strictly second largest eigenvalue of can only be achieved by at most four irreducible representations of , and we determine further the multiplicity of this eigenvalue in several special cases. As a corollary, in the case when contains neither nor we know exactly when
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A short combinatorial proof of dimension identities of Erickson and Hunziker J. Comb. Theory A (IF 0.9) Pub Date : 2024-02-29 Nishu Kumari
In a recent paper (), Erickson and Hunziker consider partitions in which the arm–leg difference is an arbitrary constant . In previous works, these partitions are called -asymmetric partitions. Regarding these partitions and their conjugates as highest weights, they prove an identity yielding an infinite family of dimension equalities between and modules. Their proof proceeds by the manipulations of
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On the deepest cycle of a random mapping J. Comb. Theory A (IF 0.9) Pub Date : 2024-02-22 Ljuben Mutafchiev, Steven Finch
Let be the set of all mappings . The corresponding graph of is a union of disjoint connected unicyclic components. We assume that each is chosen uniformly at random (i.e., with probability ). The cycle of contained within its largest component is called the one. For any , let denote the length of this cycle. In this paper, we establish the convergence in distribution of and find the limits of its expectation
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Two conjectures of Andrews, Merca and Yee on truncated theta series J. Comb. Theory A (IF 0.9) Pub Date : 2024-02-22 Shane Chern, Ernest X.W. Xia
In their study of the truncation of Euler's pentagonal number theorem, Andrews and Merca introduced a partition function to count the number of partitions of in which is the least integer that is not a part and there are more parts exceeding than there are below . In recent years, two conjectures concerning on truncated theta series were posed by Andrews, Merca, and Yee. In this paper, we prove that
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Constructing generalized Heffter arrays via near alternating sign matrices J. Comb. Theory A (IF 0.9) Pub Date : 2024-02-21 L. Mella, T. Traetta
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On the maximal number of elements pairwise generating the finite alternating group J. Comb. Theory A (IF 0.9) Pub Date : 2024-02-14 Francesco Fumagalli, Martino Garonzi, Pietro Gheri
Let be the alternating group of degree . Let be the maximal size of a subset of such that whenever and and let be the minimal size of a family of proper subgroups of whose union is . We prove that, when varies in the family of composite numbers, tends to 1 as . Moreover, we explicitly calculate for congruent to 3 modulo 18.
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A Q-polynomial structure for the Attenuated Space poset Aq(N,M) J. Comb. Theory A (IF 0.9) Pub Date : 2024-02-09 Paul Terwilliger
The goal of this article is to display a -polynomial structure for the Attenuated Space poset . The poset is briefly described as follows. Start with an -dimensional vector space over a finite field with elements. Fix an -dimensional subspace of . The vertex set of consists of the subspaces of that have zero intersection with . The partial order on is the inclusion relation. The -polynomial structure
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Most plane curves over finite fields are not blocking J. Comb. Theory A (IF 0.9) Pub Date : 2024-02-09 Shamil Asgarli, Dragos Ghioca, Chi Hoi Yip
A plane curve of degree is called if every -line in the plane meets at some -point. We prove that the proportion of blocking curves among those of degree is when and . We also show that the same conclusion holds for smooth curves under the somewhat weaker condition and . Moreover, the two events in which a random plane curve is smooth and respectively blocking are shown to be asymptotically independent
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Spectral characterization of the complete graph removing a cycle J. Comb. Theory A (IF 0.9) Pub Date : 2024-02-09 Muhuo Liu, Xiaofeng Gu, Haiying Shan, Zoran Stanić
A graph is determined by its spectrum if there is not another graph with the same spectrum. Cámara and Haemers proved that the graph , obtained from the complete graph with vertices by deleting all edges of a cycle with vertices, is determined by its spectrum for , but not for . In this paper, we show that is the unique exception for the spectral determination of .
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The divisor class group of a discrete polymatroid J. Comb. Theory A (IF 0.9) Pub Date : 2024-02-08 Jürgen Herzog, Takayuki Hibi, Somayeh Moradi, Ayesha Asloob Qureshi
In this paper we introduce toric rings of multicomplexes. We show how to compute the divisor class group and the class of the canonical module when the toric ring is normal. In the special case that the multicomplex is a discrete polymatroid, its toric ring is studied deeply for several classes of polymatroids.
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Large sum-free sets in Z5n J. Comb. Theory A (IF 0.9) Pub Date : 2024-02-02 Vsevolod F. Lev
It is well-known that for a prime and integer , the maximum possible size of a sum-free subset of the elementary abelian group is . However, the matching stability result is known for only. We consider the first open case showing that if is a sum-free subset with , then there are a subgroup of size and an element such that .
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Block-transitive 2-designs with a chain of imprimitive point-partitions J. Comb. Theory A (IF 0.9) Pub Date : 2024-02-01 Carmen Amarra, Alice Devillers, Cheryl E. Praeger
More than 30 years ago, Delandtsheer and Doyen showed that the automorphism group of a block-transitive 2-design, with blocks of size , could leave invariant a nontrivial point-partition, but only if the number of points was bounded in terms of . Since then examples have been found where there are two nontrivial point partitions, either forming a chain of partitions, or forming a grid structure on
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Large monochromatic components in colorings of complete hypergraphs J. Comb. Theory A (IF 0.9) Pub Date : 2024-02-01 Lyuben Lichev, Sammy Luo
Gyárfás famously showed that in every r-coloring of the edges of the complete graph Kn, there is a monochromatic connected component with at least nr−1 vertices. A recent line of study by Conlon, Tyomkyn, and the second author addresses the analogous question about monochromatic connected components with many edges. In this paper, we study a generalization of these questions for k-uniform hypergraphs
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A study on free roots of Borcherds-Kac-Moody Lie superalgebras J. Comb. Theory A (IF 0.9) Pub Date : 2024-01-25 Shushma Rani, G. Arunkumar
Consider a Borcherds-Kac-Moody Lie superalgebra, denoted as g, associated with the graph G. This Lie superalgebra is constructed from a free Lie superalgebra by introducing three sets of relations on its generators: (1) Chevalley relations, (2) Serre relations, and (3) The commutation relations derived from the graph G. The Chevalley relations lead to a triangular decomposition of g as g=n+⊕h⊕n−, where
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Further refinements of Wilf-equivalence for patterns of length 4 J. Comb. Theory A (IF 0.9) Pub Date : 2024-01-25 Robin D.P. Zhou, Yongchun Zang, Sherry H.F. Yan
In this paper, we construct a bijection between 3142-avoiding permutations and 3241-avoiding permutations which proves the equidistribution of five classical set-valued statistics. Our bijection also enables us to establish a bijection between 3142-avoiding permutations and 4132-avoiding permutations, and a bijection between 2413-avoiding permutations and 1423-avoiding permutations, both of which preserve
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New results on orthogonal arrays OA(3,5,4n + 2) J. Comb. Theory A (IF 0.9) Pub Date : 2024-01-24 Dongliang Li, Haitao Cao
An orthogonal array of index unity, order v, degree 5 and strength 3, or an OA(3,5,v) in short, is a 5×v3 array on v symbols and in every 3×v3 subarray, each 3-tuple column vector occurs exactly once. The existence of an OA(3,5,4n+2) is still open except for few known infinite classes of n. In this paper, we introduce a new combinatorial structure called three dimensions orthogonal complete large sets
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q-Supercongruences from Jackson's ϕ78 summation and Watson's ϕ78 transformation J. Comb. Theory A (IF 0.9) Pub Date : 2024-01-12 Chuanan Wei
q-Supercongruences modulo the fifth and sixth powers of a cyclotomic polynomial are very rare in the literature. In this paper, we establish some q-supercongruences modulo the fifth and sixth powers of a cyclotomic polynomial in terms of Jackson's ϕ78 summation, Watson's ϕ78 transformation, the creative microscoping method recently introduced by Guo and Zudilin, and the Chinese remainder theorem for
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Nowhere-zero 3-flows in Cayley graphs on supersolvable groups J. Comb. Theory A (IF 0.9) Pub Date : 2023-12-28 Junyang Zhang, Sanming Zhou
Tutte's 3-flow conjecture asserts that every 4-edge-connected graph admits a nowhere-zero 3-flow. We prove that this conjecture is true for every Cayley graph of valency at least four on any supersolvable group with a noncyclic Sylow 2-subgroup and every Cayley graph of valency at least four on any group whose derived subgroup is of square-free order.
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Algebraic approach to the completeness problem for (k,n)-arcs in planes over finite fields J. Comb. Theory A (IF 0.9) Pub Date : 2023-12-13 Gábor Korchmáros, Gábor P. Nagy, Tamás Szőnyi
In a projective plane over a finite field, complete (k,n)-arcs with few characters are rare but interesting objects with several applications to finite geometry and coding theory. Since almost all known examples are large, the construction of small ones, with k close to the order of the plane, is considered a hard problem. A natural candidate to be a small (k,n)-arc with few characters is the set Ω(C)
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Neighbour-transitive codes in Kneser graphs J. Comb. Theory A (IF 0.9) Pub Date : 2023-12-14 Dean Crnković, Daniel R. Hawtin, Nina Mostarac, Andrea Švob
A code C is a subset of the vertex set of a graph and C is s-neighbour-transitive if its automorphism group Aut(C) acts transitively on each of the first s+1 parts C0,C1,…,Cs of the distance partition {C=C0,C1,…,Cρ}, where ρ is the covering radius of C. While codes have traditionally been studied in the Hamming and Johnson graphs, we consider here codes in the Kneser graphs. Let Ω be the underlying
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Chiral polytopes whose smallest regular cover is a polytope J. Comb. Theory A (IF 0.9) Pub Date : 2023-12-08 Gabe Cunningham
We give a criterion for when the smallest regular cover of a chiral polytope P is itself a polytope, using only information about the facets and vertex-figures of P.
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Asymptotics for real monotone double Hurwitz numbers J. Comb. Theory A (IF 0.9) Pub Date : 2023-12-08 Yanqiao Ding, Qinhao He
In recent years, monotone double Hurwitz numbers were introduced as a naturally combinatorial modification of double Hurwitz numbers. Monotone double Hurwitz numbers share many structural properties with their classical counterparts, such as piecewise polynomiality, while the quantitative properties of these two numbers are quite different. We consider real analogues of monotone double Hurwitz numbers
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The method of constant terms and k-colored generalized Frobenius partitions J. Comb. Theory A (IF 0.9) Pub Date : 2023-12-01 Su-Ping Cui, Nancy S.S. Gu, Dazhao Tang
In his 1984 AMS memoir, Andrews introduced the family of k-colored generalized Frobenius partition functions. For any positive integer k, let cϕk(n) denote the number of k-colored generalized Frobenius partitions of n. Among many other things, Andrews proved that for any n≥0, cϕ2(5n+3)≡0(mod5). Since then, many scholars subsequently considered congruence properties of various k-colored generalized
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Monochromatic arithmetic progressions in automatic sequences with group structure J. Comb. Theory A (IF 0.9) Pub Date : 2023-12-01 Ibai Aedo, Uwe Grimm, Neil Mañibo, Yasushi Nagai, Petra Staynova
We determine asymptotic growth rates for lengths of monochromatic arithmetic progressions in certain automatic sequences. In particular, we look at (one-sided) fixed points of aperiodic, primitive, bijective substitutions and spin substitutions, which are generalisations of the Thue–Morse and Rudin–Shapiro substitutions, respectively. For such infinite words, we show that there exists a subsequence
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The second-order football-pool problem and the optimal rate of generalized-covering codes J. Comb. Theory A (IF 0.9) Pub Date : 2023-11-28 Dor Elimelech, Moshe Schwartz
The goal of the classic football-pool problem is to determine how many lottery tickets are to be bought in order to guarantee at least n−r correct guesses out of a sequence of n games played. We study a generalized (second-order) version of this problem, in which any of these n games consists of two sub-games. The second-order version of the football-pool problem is formulated using the notion of
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MacMahon's partition analysis XIV: Partitions with n copies of n J. Comb. Theory A (IF 0.9) Pub Date : 2023-11-28 George E. Andrews, Peter Paule
We apply the methods of partition analysis to partitions with n copies of n. This allows us to obtain multivariable generating functions related to classical Rogers-Ramanujan type identities. Also, partitions with n copies of n are extended to partition diamonds yielding numerous new results including a natural connection to overpartitions and a variety of partition congruences.
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Singleton-type bounds for list-decoding and list-recovery, and related results J. Comb. Theory A (IF 0.9) Pub Date : 2023-11-28 Eitan Goldberg, Chong Shangguan, Itzhak Tamo
List-decoding and list-recovery are important generalizations of unique decoding and receive considerable attention over the years. We study the optimal trade-off among the list-decoding (resp. list-recovery) radius, the list size, and the code rate, when the list size is constant and the alphabet size is large (both compared with the code length). We prove a new Singleton-type bound for list-decoding
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Matroid Horn functions J. Comb. Theory A (IF 0.9) Pub Date : 2023-11-24 Kristóf Bérczi, Endre Boros, Kazuhisa Makino
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
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The Clebsch–Gordan coefficients of U(sl2) and the Terwilliger algebras of Johnson graphs J. Comb. Theory A (IF 0.9) Pub Date : 2023-11-16 Hau-Wen Huang
The universal enveloping algebra U(sl2) of sl2 is a unital associative algebra over C generated by E,F,H subject to the relations[H,E]=2E,[H,F]=−2F,[E,F]=H. The elementΛ=EF+FE+H22 is called the Casimir element of U(sl2). Let Δ:U(sl2)→U(sl2)⊗U(sl2) denote the comultiplication of U(sl2). The universal Hahn algebra H is a unital associative algebra over C generated by A,B,C and the relations assert that
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Some refinements of Stanley's shuffle theorem J. Comb. Theory A (IF 0.9) Pub Date : 2023-11-17 Kathy Q. Ji, Dax T.X. Zhang
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.
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A modular approach to Andrews-Beck partition statistics J. Comb. Theory A (IF 0.9) Pub Date : 2023-11-15 Renrong Mao
Andrews recently provided a q-series proof of congruences for NT(m,k,n), the total number of parts in the partitions of n with rank congruent to m modulo k. Motivated by Andrews' works, Chern obtain congruences for Mω(m,k,n) which denotes the total number of ones in the partition of n with crank congruent to m modulo k. In this paper, we focus on the modular approach to these new partition statistics
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A bivariate Q-polynomial structure for the non-binary Johnson scheme J. Comb. Theory A (IF 0.9) Pub Date : 2023-10-24 Nicolas Crampé, Luc Vinet, Meri Zaimi, Xiaohong Zhang
The notion of multivariate P- and Q-polynomial association scheme has been introduced recently, generalizing the well-known univariate case. 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 P-polynomial association scheme. We show here that it is also a bivariate Q-polynomial association
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Non-expansive matrix number systems with bases similar to certain Jordan blocks J. Comb. Theory A (IF 0.9) Pub Date : 2023-10-19 Joshua W. Caldwell, Kevin G. Hare, Tomáš Vávra
We study representations of integral vectors in a number system with a matrix base M and vector digits. We focus on the case when M is equal or similar to Jn, the Jordan block with eigenvalue 1 and dimension n. If M=J2, we classify all digit sets of size two allowing representation for all of Z2. For M=Jn with n≥3, we show that a digit set of size three suffice to represent all of Zn. For bases M similar
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On some double Nahm sums of Zagier J. Comb. Theory A (IF 0.9) Pub Date : 2023-10-11 Zhineng Cao, Hjalmar Rosengren, Liuquan Wang
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 q-series proof for the tenth example. We prove that the fifth and the tenth examples are in fact equivalent. Then we give a q-series proof for the fifth example, which confirms a recent conjecture of Wang. This also serves as the first
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Union-closed sets and Horn Boolean functions J. Comb. Theory A (IF 0.9) Pub Date : 2023-10-11 Vadim Lozin, Viktor Zamaraev
A family F of sets is union-closed if the union of any two sets from F belongs to F. The union-closed sets conjecture states that if F is a finite union-closed family of finite sets, then there is an element that belongs to at least half of the sets in F. The conjecture has several equivalent formulations in terms of other combinatorial structures such as lattices and graphs. In its whole generality
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Partitioning into common independent sets via relaxing strongly base orderability J. Comb. Theory A (IF 0.9) Pub Date : 2023-09-22 Kristóf Bérczi, Tamás Schwarcz
The problem of covering the ground set of two matroids by a minimum number of common independent sets is notoriously hard even in very restricted settings, i.e. when the goal is to decide if two common independent sets suffice or not. Nevertheless, as the problem generalizes several long-standing open questions, identifying tractable cases is of particular interest. Strongly base orderable matroids
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Two-geodesic transitive graphs of order pn with n ≤ 3 J. Comb. Theory A (IF 0.9) Pub Date : 2023-09-18 Jun-Jie Huang, Yan-Quan Feng, Jin-Xin Zhou, Fu-Gang Yin
A vertex triple (u,v,w) of a graph is called a 2-geodesic if v is adjacent to both u and w and u is not adjacent to w. A graph is said to be 2-geodesic transitive 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 pn is given for each prime p and n≤3. It turns out that all such graphs consist of three
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A bijection for length-5 patterns in permutations J. Comb. Theory A (IF 0.9) Pub Date : 2023-09-18 Joanna N. Chen, Zhicong Lin
A bijection which preserves five classical set-valued permutation statistics between (31245,32145,31254,32154)-avoiding permutations and (31425,32415,31524,32514)-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
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Trianguloids and triangulations of root polytopes J. Comb. Theory A (IF 0.9) Pub Date : 2023-09-11 Pavel Galashin, Gleb Nenashev, Alexander Postnikov
Triangulations of a product of two simplices and, more generally, of root polytopes are closely related to Gelfand-Kapranov-Zelevinsky's theory of discriminants, to tropical geometry, tropical oriented matroids, and to generalized permutohedra. We introduce a new approach to these objects, identifying a triangulation of a root polytope with a certain bijection between lattice points of two generalized
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Real equiangular lines in dimension 18 and the Jacobi identity for complementary subgraphs J. Comb. Theory A (IF 0.9) Pub Date : 2023-09-08 Gary R.W. Greaves, Jeven Syatriadi
We show that the maximum cardinality of an equiangular line system in R18 is at most 59. Our proof includes a novel application of the Jacobi identity for complementary subgraphs. In particular, we show that there does not exist a graph whose adjacency matrix has characteristic polynomial (x−22)(x−2)42(x+6)15(x+8)2.
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Singleton mesh patterns in multidimensional permutations J. Comb. Theory A (IF 0.9) Pub Date : 2023-09-08 Sergey Avgustinovich, Sergey Kitaev, Jeffrey Liese, Vladimir Potapov, Anna Taranenko
This paper introduces the notion of mesh patterns in multidimensional permutations and initiates a systematic study of singleton mesh patterns (SMPs), which are multidimensional mesh patterns of length 1. A pattern is avoidable if there exist arbitrarily large permutations that do not contain it. As our main result, we give a complete characterization of avoidable SMPs using an invariant of a pattern
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Improved Elekes-Szabó type estimates using proximity J. Comb. Theory A (IF 0.9) Pub Date : 2023-09-07 Jozsef Solymosi, Joshua Zahl
We prove a new Elekes-Szabó type estimate on the size of the intersection of a Cartesian product A×B×C with an algebraic surface {f=0} over the reals. In particular, if A,B,C are sets of N real numbers and f is a trivariate polynomial, then either f has a special form that encodes additive group structure (for example, f(x,y,x)=x+y−z), or A×B×C∩{f=0} has cardinality O(N12/7). This is an improvement
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Constructing uniform 2-factorizations via row-sum matrices: Solutions to the Hamilton-Waterloo problem J. Comb. Theory A (IF 0.9) Pub Date : 2023-09-01 A.C. Burgess, P. Danziger, A. Pastine, T. Traetta
In this paper, we formally introduce the concept of a row-sum matrix over an arbitrary group G. When G is cyclic, these types of matrices have been widely used to build uniform 2-factorizations of small Cayley graphs (or, Cayley subgraphs of blown-up cycles), which themselves factorize complete (equipartite) graphs. Here, we construct row-sum matrices over a class of non-abelian groups, the generalized
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Weighted Subspace Designs from q-Polymatroids J. Comb. Theory A (IF 0.9) Pub Date : 2023-08-22 Eimear Byrne, Michela Ceria, Sorina Ionica, Relinde Jurrius
The Assmus-Mattson Theorem gives a way to identify block designs arising from codes. This result was broadened to matroids and weighted designs by Britz et al. in 2009. In this work we present a further two-fold generalisation: first from matroids to polymatroids and also from sets to vector spaces. To achieve this, we study the characteristic polynomial of a q-polymatroid and outline several of its
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On the polymatroid Tutte polynomial J. Comb. Theory A (IF 0.9) Pub Date : 2023-08-16 Xiaxia Guan, Weiling Yang, Xian'an Jin
The Tutte polynomial is a well-studied invariant of matroids. The polymatroid Tutte polynomial TP(x,y), introduced by Bernardi, Kálmán, and Postnikov, is an extension of the classical Tutte polynomial from matroids to polymatroids P. In this paper, we first prove that TP(x,t) and TP(t,y) are interpolating for any fixed real number t≥1, and then we study the coefficients of high-order terms in TP(x
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The full automorphism groups of general position graphs J. Comb. Theory A (IF 0.9) Pub Date : 2023-08-14 Junyao Pan
Let S be a non-empty finite set. A flag of S is a set f of non-empty proper subsets of S such that X⊆Y or Y⊆X for all X,Y∈f. The set {|X|:X∈f} is called the type of f. Two flags f and f′ are in general position with respect to S if X∩Y=∅ or X∪Y=S for all X∈f and Y∈f′. For a fixed type T, Klaus Metsch defined the general position graph Γ(S,T) whose vertices are the flags of S of type T with two vertices
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Proof of Dilks' bijectivity conjecture on Baxter permutations J. Comb. Theory A (IF 0.9) Pub Date : 2023-08-07 Zhicong Lin, Jing Liu
Baxter permutations originally arose in studying common fixed points of two commuting continuous functions. In 2015, Dilks proposed a conjectured bijection between Baxter permutations and non-intersecting triples of lattice paths in terms of inverse descent bottoms, descent positions and inverse descent tops. We prove this bijectivity conjecture by investigating its connection with the Françon–Viennot
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Spinning switches on a wreath product J. Comb. Theory A (IF 0.9) Pub Date : 2023-08-03 Peter Kagey
We classify an algebraic phenomenon on several families of wreath products that can be seen as coming from a generalization of a puzzle about switches on the corners of a spinning table. Such puzzles have been written about and generalized since they were first popularized by Martin Gardner in 1979. In this paper, we build upon a paper of Bar Yehuda, Etzion, and Moran, a paper of Ehrenborg and Skinner
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Hook length and symplectic content in partitions J. Comb. Theory A (IF 0.9) Pub Date : 2023-07-31 T. Amdeberhan, G.E. Andrews, C. Ballantine
The dimension of an irreducible representation of GL(n,C), Sp(2n), or SO(n) is given by the respective hook length and content formulas for the corresponding partition. The first author, inspired by the Nekrasov-Okounkov formula, conjectured combinatorial interpretations of analogous expressions involving hook lengths and symplectic/orthogonal contents. We prove special cases of these conjectures.
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A family of diameter perfect constant-weight codes from Steiner systems J. Comb. Theory A (IF 0.9) Pub Date : 2023-07-31 Minjia Shi, Yuhong Xia, Denis S. Krotov
If S is a transitive metric space, then |C|⋅|A|≤|S| for any distance-d code C and a set A, “anticode”, of diameter less than d. For every Steiner S(t,k,n) system S, we show the existence of a q-ary constant-weight code C of length n, weight k (or n−k), and distance d=2k−t+1 (respectively, d=n−t+1) and an anticode A of diameter d−1 such that the pair (C,A) attains the code–anticode bound and the supports
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Modifications of hyperplane arrangements J. Comb. Theory A (IF 0.9) Pub Date : 2023-07-31 Houshan Fu, Suijie Wang
This paper is concerned with five kinds of modification of hyperplane arrangements, including elementary lift, parallel translation, coning, one-element extension and restriction to a hyperplane. We show that the combinatorial classification of all hyperplane arrangements of each kind of modification will be characterized by the intersection lattice of the discriminantal or adjoint arrangement. Based
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Linear configurations containing 4-term arithmetic progressions are uncommon J. Comb. Theory A (IF 0.9) Pub Date : 2023-07-26 Leo Versteegen
A linear configuration is said to be common in an Abelian group G if every 2-coloring of G yields at least the number of monochromatic instances of a randomly chosen coloring. Saad and Wolf asked whether, analogously to a result by Jagger, Šťovíček and Thomason in graph theory, every configuration containing a 4-term arithmetic progression is uncommon. We prove this in Fpn for p≥5 and large n and in
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Hermitian matrices of roots of unity and their characteristic polynomials J. Comb. Theory A (IF 0.9) Pub Date : 2023-07-20
We investigate spectral conditions on Hermitian matrices of roots of unity. Our main results are conjecturally sharp upper bounds on the number of residue classes of the characteristic polynomial of such matrices modulo ideals generated by powers of (1−ζ), where ζ is a root of unity. We also prove a generalisation of a classical result of Harary and Schwenk about a relation for traces of powers of
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Sum formulas for Schur multiple zeta values J. Comb. Theory A (IF 0.9) Pub Date : 2023-07-18
In this paper, we study sum formulas for Schur multiple zeta values and give a generalization of the sum formulas for multiple zeta(-star) values. We show that for ribbons of certain types, the sum of Schur multiple zeta values over all admissible Young tableaux of this shape evaluates to a rational multiple of the Riemann zeta value. For arbitrary ribbons with n corners, we show that such a sum can
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A product version of the Hilton-Milner theorem J. Comb. Theory A (IF 0.9) Pub Date : 2023-07-17
Two families F,G of k-subsets of {1,2,…,n} are called non-trivial cross-intersecting if F∩G≠∅ for all F∈F,G∈G and ∩{F:F∈F}=∅=∩{G:G∈G}. In the present paper, we determine the maximum product of the sizes of two non-trivial cross-intersecting families of k-subsets of {1,2,…,n} for n≥4k, k≥8, which is a product version of the classical Hilton-Milner Theorem.
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Linked partition ideals and a family of quadruple summations J. Comb. Theory A (IF 0.9) Pub Date : 2023-07-17
Recently, 4-regular partitions into distinct parts were connected with a family of overpartitions. In this paper, we provide a uniform extension of two relations due to Andrews for the two types of partitions. Such an extension is made possible with recourse to a new trivariate Rogers–Ramanujan type identity, which concerns a family of quadruple summations appearing as generating functions for the
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The first infinite family of orthogonal Steiner systems S(3,5,v) J. Comb. Theory A (IF 0.9) Pub Date : 2023-07-04 Qianqian Yan, Junling Zhou
The research on orthogonal Steiner systems S(t,k,v) was initiated in 1968. For (t,k)∈{(2,3),(3,4)}, this corresponds to orthogonal Steiner triple systems (STSs) and Steiner quadruple systems (SQSs), respectively. The existence problem of a pair of orthogonal STSs or SQSs was settled completely thirty years ago. However, for Steiner systems with t≥3 and k≥5, only two small examples of orthogonal pairs