Let S={x1,…,xn} be a gcd-closed set (i.e. (xi,xj)∈S for all 1≤i,j≤n). In 2002, Hong proposed the divisibility problem of characterizing all gcd-closed sets S with |S|≥4 such that the GCD matrix (S) divides the LCM matrix [S] in the ring Mn(Z). For x∈S, let GS(x):={z∈S:z

A multidimensional nonnegative matrix is called polystochastic if the sum of its entries over each line is equal to 1. The permanent of a multidimensional matrix is the sum of products of entries over all diagonals. We prove that if d is even, then the permanent of a d-dimensional polystochastic matrix of order 4 is positive, and for odd d, we give a complete characterization of d-dimensional polystochastic

Motivated by Kitaev and Zhang's recent work on non-overlapping ascents in stack-sortable permutations and Dumont's permutation interpretation of the Jacobi elliptic functions, we investigate some parity statistics on restricted permutations. Some new related bijections are constructed and two refinements of the generating function for descents over 321-avoiding permutations due to Barnabei, Bonetti

Divisible design graphs were introduced in 2011 by Haemers, Kharaghani and Meulenberg. In this paper, we introduce the notion of q-analogs of divisible design graphs and show that all q-analogs of divisible design graphs come from spreads, and are actually q-analogs of strongly regular graphs.

For each odd prime power q, we describe a class of rational functions f(X)∈Fq(X) with the following unusual property: for every odd j, the function induced by f(X) on Fqj∪{∞} is 2-to-1. We also show that, among all known rational functions f(X)∈Fq(X) which are 2-to-1 on Fqj∪{∞} for infinitely many j, our new functions are the only ones which cannot be written as compositions of rational functions of

Given a positive integer n, consider a permutation of n objects chosen uniformly at random. In this permutation, we collect maximal subsequences consisting of consecutive numbers arranged in ascending order called blocks. Each block is then merged, and after all merges, the elements of this new set are relabeled from 1 to the current number of elements. We continue to permute and merge this new set

Let Γ be a Cayley graph on a finite group G, and let NAut(Γ)(R(G)) be the normalizer of R(G) (the right regular representation of G) in the full automorphism group Aut(Γ) of Γ. We say that Γ is a normal Cayley graph on G if NAut(Γ)(R(G))=Aut(Γ), and that Γ is a normal edge-transitive Cayley graph on G if NAut(Γ)(R(G)) acts transitively on the edge set of Γ. In 1999, Praeger proved that every connected

In this article, we discuss finite versions of Euler's pentagonal number identity, the Rogers-Ramanujan identities and present new proofs of the finite versions of the Andrews-Gordon identity and the Bressoud identity. We also investigate the finite version of Garvan's generalizations of Dyson's rank and discover a new one-variable extension of the Andrews-Gordon identity.

An association scheme is called amorphic if every possible fusion of relations gives rise to a fusion scheme. We call a pair of relations fusing if fusing that pair gives rise to a fusion scheme. We define the fusing-relations graph on the set of relations, where a pair forms an edge if it fuses. We show that if the fusing-relations graph is connected but not a path, then the association scheme is

The purpose of this paper is two-fold. First, we characterize the existence of binary self-orthogonal codes meeting the Griesmer bound by employing the Solomon-Stiffler codes. As a result, we reduce a problem with an infinite number of cases to a finite number of cases. Second, we develop a general method to prove the nonexistence of some binary self-orthogonal codes by considering the residual code

In this paper, we find distributions of the left-to-right maxima, right-to-left maxima, left-to-right minima and right-to-left-minima statistics on up-down and down-up permutations of even and odd lengths. We recover and generalize a result by Carlitz and Scoville, obtained in 1975, stating that the distribution of left-to-right maxima on down-up permutations of even length is given by (sec⁡(t))q.

Bressoud introduced the partition function B(α1,…,αλ;η,k,r;n), which counts the number of partitions with certain difference conditions. Bressoud posed a conjecture on the generating function for the partition function B(α1,…,αλ;η,k,r;n) in multi-summation form. In this article, we introduce a bijection related to Bressoud's conjecture. As an application, we give the proof of a companion to the Göllnitz-Gordon

De Bruijn tori, or perfect maps, are two-dimensional periodic arrays of letters from a finite alphabet, where each possible pattern of shape (m,n) appears exactly once in a single period. While the existence of certain de Bruijn tori, such as square tori with odd m=n∈{3,5,7} and even alphabet sizes, remains unresolved, sub-perfect maps are often sufficient in applications like positional coding. These

The Clebsch–Gordan coefficients of U(sl2) are expressible in terms of Hahn polynomials. The phenomenon can be explained by an algebra homomorphism ♮ from the universal Hahn algebra H into U(sl2)⊗U(sl2). Let Ω denote a finite set of size D and 2Ω denote the power set of Ω. It is generally known that C2Ω supports a U(sl2)-module. Let k denote an integer with 0≤k≤D and fix a k-element subset x0 of Ω.

Intersecting codes are linear codes where every two nonzero codewords have non-trivially intersecting support. In this article we expand on the theory of this family of codes, by showing that nondegenerate intersecting codes correspond to sets of points (with multiplicities) in a projective space that are not contained in two hyperplanes. This correspondence allows the use of geometric arguments to

Separable elements in Weyl groups are generalizations of the well-known class of separable permutations in symmetric groups. Gaetz and Gao showed that for any pair (X,Y) of subsets of the symmetric group Sn, the multiplication map X×Y→Sn is a splitting (i.e., a length-additive bijection) of Sn if and only if X is the generalized quotient of Y and Y is a principal lower order ideal in the right weak

Recently, Leemans and Stokes constructed an infinite family of incidence geometries admitting trialities but no dualities from the groups PSL(2,q) (where q=p3n with p a prime and n>0 a positive integer). Unfortunately, these geometries are not flag transitive. In this paper, we work with the Suzuki groups Sz(q), where q=22e+1 with e a positive integer and 2e+1 is divisible by 3. For any odd integer

The classification problem of P- and Q-polynomial association schemes has been one of the central problems in algebraic combinatorics. Generalizing the concept of P- and Q-polynomial association schemes to multivariate cases, namely to consider higher rank P- and Q-polynomial association schemes, has been tried by some authors, but it seems that so far there were neither very well-established definitions

Let V be a finite dimensional vector space over a finite field. Suppose that F1, F2, …, Fr are r-cross t-intersecting families of k-subspaces of V. In this paper, we determine the extremal structure when ∏i=1r|Fi| is maximum under the condition that dim⁡(⋂F∈FiF)

Cameron-Liebler sets of generators in polar spaces were introduced a few years ago as natural generalisations of the Cameron-Liebler sets of subspaces in projective spaces. In this article we present the first two constructions of non-trivial Cameron-Liebler sets of generators in polar spaces. Also regular m-ovoids of k-spaces are introduced as a generalization of m-ovoids of polar spaces. They are

We obtain a polynomial criterion for a set to have a small doubling in terms of the common energy of its subsets.

We show that the Terwilliger algebra of a quasi-thin association scheme over a field is always a quasi-hereditary cellular algebra in the sense of Cline-Parshall-Scott and of Graham-Lehrer, respectively, and that the basic algebra of the Terwilliger algebra is the dual extension of a star with all arrows pointing to its center if the field has characteristic 2. Thus many homological and representation-theoretic

We construct simple geometric operations on faces of the Cayley sum of two polytopes. These operations can be thought of as convex geometric counterparts of divided difference operators in Schubert calculus. We show that these operations give a uniform construction of Knutson–Miller mitosis in the type A and Fujita mitosis in the type C on Kogan faces of Gelfand–Zetlin polytopes.

In 1920, Percy Alexander MacMahon defined the partition generating functionsAk(q):=∑0

By considering the parity of the degrees and levels of nodes in increasing trees, a new combinatorial interpretation for the coefficients of the Taylor expansions of the Jacobi elliptic functions is found. As one application of this new interpretation, a conjecture of Ma–Mansour–Wang–Yeh is solved. Unifying the concepts of increasing trees and plane trees, Lin–Ma–Ma–Zhou introduced weakly increasing

This paper concentrates on recursive constructions for 2-designs over finite fields. In 1998, Itoh presented a powerful recursive construction: for certain index λ, if there exists a Singer cycle invariant 2-(l,3,λ)q design, then there also exists an SL(m,ql) invariant 2-(ml,3,λ)q design for all integers m≥3. We investigate the GL(m,ql)-incidence matrix between 2-subspaces and k-subspaces of GF(q)ml

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 c∈(0,1] and all positive integers N, subsets of density at least c in {0,1,…,N−1}n contain hypercubes of arbitrarily large dimension as n grows.

Let k≥2 be an integer and let A be a set of nonnegative integers. For a k-tuple of positive integers λ_=(λ1,…,λk) with 1≤λ1<λ2<⋯<λk, we define the additive representation function RA,λ_(n)=|{(a1,…,ak)∈Ak:λ1a1+⋯+λkak=n}|. For k=2, Moser constructed a set A of nonnegative integers such that RA,λ_(n)=1 holds for every nonnegative integer n. In this paper we characterize all the k-tuples λ_ and the sets

A pair (st1,st2) of permutation statistics is said to be r-Euler-Mahonian if (st1,st2) and (rdes, rmaj) are equidistributed over the set Sn of all permutations of {1,2,…,n}, where rdes denotes the r-descent number and rmaj denotes the r-major index introduced by Rawlings. The main objective of this paper is to prove that (excr,denr) and (rdes, rmaj) are equidistributed over Sn, thereby confirming a

For a linear Hamming metric code of length n over a finite field, the number of distinct weights of its codewords is at most n. 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

Let N, P 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 [14] conjectured that the equation (⁎)x2=22a+2p2n−2a+2pm+n+1, p∈P,x,z,m,n∈N has only one solution (p,x,a,m,n)=(5,49,3,2,1). This is a far from solved problem that has been poorly known for so long. In this paper, using some elementary

We study the diametric problem (i.e., optimal anticodes) in the space of permutations under the Ulam distance. That is, let Sn denote the set of permutations on n symbols, and for each σ,τ∈Sn, 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

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

Motivated by complexity questions in integer programming, this paper aims to contribute to the understanding of combinatorial properties of integer matrices of row rank r and with bounded subdeterminants. In particular, we study the column number question for integer matrices whose every r×r minor is non-zero and bounded by a fixed constant Δ in absolute value. Approaching the problem in two different

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 M to be a Cayley extension of K if the facets of M are isomorphic to K and if some subgroup of the automorphism group of M acts regularly on the facets of M. We show that many natural extensions in the literature on

Let (G,+,0) be a finite abelian group and let ηN(G) be the smallest integer ℓ such that every sequence over G∖{0} of length ℓ has two joint short minimal zero-sum subsequences. In 2013, Gao et al. obtained that ηN(Cn⊕Cn)=3n+1 for every n≥2 and solved the corresponding inverse problem for groups Cp⊕Cp, where p is a prime. In this paper, we determine the precise value of ηN(G) for all finite abelian

In 1982, Cameron and Liebler investigated certain special sets of lines in PG(3,q), and gave several equivalent characterizations. Due to their interesting geometric and algebraic properties, these Cameron-Liebler line classes got much attention. Several generalizations and variants have been considered in the literature, the main directions being a variation of the dimensions of the involved spaces

Transmit a codeword ▪, that belongs to an (ℓ−1)-deletion-correcting code of length n, over a t-deletion channel for some 1≤ℓ≤t

In this paper, we provide a complete classification of 2-(v,k,2) designs admitting a flag-transitive automorphism group of affine type with the only exception of the semilinear 1-dimensional group. Alongside this analysis, we provide a construction of seven new families of such flag-transitive 2-designs, one of them infinite, and some of them involving remarkable objects such as t-spreads, translation

Let Dn be the set of partitions of n into distinct parts, and srp(λ) be the sum of reciprocals of the parts of the partition λ. We show that as n→∞,E[srp(λ):λ∈Dn]∼log⁡(3n)4andVar[srp(λ):λ∈Dn]∼π224. Moreover, for Pn, the set of ordinary partitions of n, we show that as n→∞,E[srp(λ):λ∈Pn]∼πn6andVar[srp(λ):λ∈Pn]∼π215n. To prove these asymptotic formulas in a uniform manner, we derive a general asymptotic

Two families A and B are cross-intersecting if A∩B≠∅ for any A∈A and B∈B. We call t families A1,A2,…,At pairwise cross-intersecting families if Ai and Aj are cross-intersecting for 1≤i

We show that the dominance complex D(G) of a graph G coincides with the combinatorial Alexander dual of the neighborhood complex N(G‾) of the complement of G. Using this, we obtain a relation between the chromatic number χ(G) of G and the homology group of D(G). We also obtain several known results related to dominance complexes from well-known facts of neighborhood complexes. After that, we suggest

Motivated by a classical result of Graver and Jurkat (1973) and Graham, Li, and Li (1980) in combinatorial design theory, which states that the permutations of t-(v,k) minimal trades generate the vector space of all t-(v,k) trades, we investigate the vector space spanned by permutations of an arbitrary trade. We prove that this vector space possesses a decomposition as a direct sum of subspaces formed

We use techniques from algebraic and extremal combinatorics to derive upper bounds on the number of independent sets in several (hyper)graphs arising from finite geometry. In this way, we obtain asymptotically sharp upper bounds for partial ovoids and EKR-sets of flags in polar spaces, line spreads in PG(2r−1,q) and plane spreads in PG(5,q), and caps in PG(3,q). The latter result extends work due to

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

For a given n, what is the smallest number k such that every sequence of length n is determined by the multiset of all its k-subsequences? This is called the k-deck problem for sequence reconstruction, and has been generalized to the two-dimensional case – reconstruction of n×n-matrices from submatrices. Previous works show that the smallest k is at most O(n12) for sequences and at most O(n23) for

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

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

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.

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.

We investigate locally n×n grid graphs, that is, graphs in which the neighbourhood of any vertex is the Cartesian product of two complete graphs on n 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 2n by previous work of Blokhuis and Brouwer. We show that

Endowed with the binary operation of set addition, the family Pfin,0(N) of all finite subsets of N containing 0 forms a monoid, with the singleton {0} as its neutral element.

Let A1,A2,…,Am be families of k-element subsets of a n-element set. We call them cross-t-intersecting if |Ai∩Aj|≥t for any Ai∈Ai and Aj∈Aj with i≠j. In this paper we will prove that, for n≥2k−t+1, if A1,A2,…,Am are non-empty cross-t-intersecting families, then∑1≤i≤m|Ai|≤max⁡{(nk)−∑1≤i≤t−1(ki)(n−kk−i)+m−1,mM(n,k,t)}, where M(n,k,t) is the size of the maximum t-intersecting family of ([n]k). Moreover

We study the set of intersection sizes of a k-dimensional affine subspace and a point set of size m∈[0,2n] of the n-dimensional binary affine space AG(n,2). Following the theme of Erdős, Füredi, Rothschild and T. Sós, we partially determine which local densities in k-dimensional affine subspaces are unavoidable in all m-element point sets in the n-dimensional affine space.

Let Γ=Γ(n,m) denote the Doob graph formed by the Cartesian product of the nth Cartesian power of the Shrikhande graph and the mth Cartesian power of the complete graph on four vertices. Let T=T(x) denote the Terwilliger algebra of Γ with respect to a fixed vertex x of Γ and let W denote an arbitrary non-thin irreducible T-module in the standard module of Γ. In (Morales and Palma, 2021 [25]), it was

Let p be a prime and let G be a finite group which is generated by the set Gp of its p-elements. We show that if G is solvable and not a p-group, then the minimal number σp(G) of proper subgroups of G whose union contains Gp is equal to 1 less than the minimal number of proper subgroups of G whose union is G. For p-solvable groups G, we always have σp(G)≥p+1. We study the case of alternating and symmetric

In 2012, Andrews and Merca investigated the truncated version of the Euler pentagonal number theorem. Their work has opened up a new study on truncated theta series and has inspired several mathematicians to work on the topic. In 2019, Merca studied the Rogers-Ramanujan functions and posed three groups of conjectures on truncated series involving the Rogers-Ramanujan functions. In this paper, we present

We study the proportion of metric matroids whose Jacobians have nontrivial p-torsion. We establish a correspondence between these Jacobians and the Fp-rational points on configuration hypersurfaces, thereby relating their proportions. By counting points over finite fields, we prove that the proportion of these Jacobians is asymptotically equivalent to 1/p.