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.

Constant weight codes (CWCs) and constant composition codes (CCCs) are two important classes of codes that have been studied extensively in both combinatorics and coding theory for nearly sixty years. In this paper we show that for all fixed odd distances, there exist near-optimal CWCs and CCCs asymptotically achieving the classic Johnson-type upper bounds.

For a positive integer m, a group G is said to admit a tournament m-semiregular representation (TmSR for short) if there exists a tournament Γ such that the automorphism group of Γ is isomorphic to G and acts semiregularly on the vertex set of Γ with m orbits. It is easy to see that every finite group of even order does not admit a TmSR for any positive integer m. The T1SR is the well-known tournament

The exact value of the separating Noether number of an arbitrary finite abelian group of rank two is determined. This is done by a detailed study of the monoid of zero-sum sequences over the group.

The tableau reconstruction problem, posed by Monks (2009), asks the following. Starting with a standard Young tableau T, a 1-minor of T is a tableau obtained by first deleting any cell of T, and then performing jeu de taquin slides to fill the resulting gap. This can be iterated to arrive at the set of k-minors of T. The problem is this: given k, what are the values of n such that every tableau of

In this paper, we establish some general expansion formulas for q-series. Three of Liu's identities motivate us to search and find such type of formulas. These expansion formulas include as special cases or limiting cases many q-identities including the q-Gauss summation formula, the q-Pfaff-Saalschütz summation formula, three of Jackson's transformation formulas and Sears' terminating ϕ34 transformation

Let and denote the permutation statistics -descent number and -excedance number, respectively. We prove that the pairs of permutation statistics and are equidistributed, where denotes the -major index defined by Don Rawlings and denotes the -Denert's statistic defined by Guo-Niu Han. When , this result reduces to the equidistribution of and , which was conjectured by Denert in 1990 and proved that

The -Onsager algebra, denoted , is defined by two generators and two relations called the -Dolan-Grady relations. Recently, Terwilliger introduced some elements of , said to be alternating. These elements are denoted

We construct an infinite family of hyperovals on the Klein quadric , even. The construction makes use of ovoids of the symplectic generalized quadrangle that is associated with an elliptic quadric which arises as solid intersection with . We also solve the isomorphism problem: we determine necessary and sufficient conditions for two hyperovals arising from the construction to be isomorphic.

Ferrers diagram rank-metric codes were introduced by Etzion and Silberstein in 2009. In their work, they proposed a conjecture on the largest dimension of a space of matrices over a finite field whose nonzero elements are supported on a given Ferrers diagram and all have rank lower bounded by a fixed positive integer . Since stated, the Etzion-Silberstein conjecture has been verified in a number of

A is a set of positions in a word such that each distinct factor has an occurrence crossing a position from the set. This definition comes from the data compression field, where the size of a smallest string attractor represents a lower bound for the output size of a large family of string compressors exploiting repetitions in words, including BWT-based and LZ-based compressors. For finite words, the

Cluster exchange groupoids are introduced by King-Qiu as an enhancement of cluster exchange graphs to study stability conditions and quadratic differentials. In this paper, we introduce the cluster exchange groupoid for any finite Coxeter-Dynkin diagram Δ and show that its fundamental group is isomorphic to the corresponding braid group associated with Δ.

In this paper, we present a study on , which are polygons created by connecting unit squares along their edges. Specifically, we focus on a related concept called a , which is a path on a lattice in traced along the boundaries of a column convex polyomino where the lower edge is on the -axis. To explore new variations of bargraphs, we introduce the notion of , which incorporate an additional restriction

In this paper, we refine a result of Andrews and Merca on truncated pentagonal number series. Subsequently, we establish some positivity results involving Andrews–Gordon–Bressoud identities and -regular partitions. In particular, we prove several conjectures of Merca and Krattenthaler–Merca–Radu on truncated pentagonal number series.

The difference graph of a finite group is the difference of the enhanced power graph of and the power graph of , where all isolated vertices are removed. In this paper we study the connectedness and perfectness of with respect to various properties of the underlying group . We also find several connections between the difference graph of and the Gruenberg-Kegel graph of . We also examine the operation

In this note we show that a flag-transitive automorphism group of a non-trivial 2- design with is not of product action type. In conclusion, is a primitive group of affine or almost simple type.

A subset of a finite transitive group is if any two elements of agree on an element of Ω. The of is the number where and is the stabilizer of in . It is known that if is an imprimitive group of degree a product of two odd primes admitting a block of size or two complete block systems, whose blocks are of size , then .

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

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

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

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

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

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.

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

A plane curve C⊂P2 of degree d is called blocking if every Fq-line in the plane meets C at some Fq-point. We prove that the proportion of blocking curves among those of degree d is o(1) when d≥2q−1 and q→∞. We also show that the same conclusion holds for smooth curves under the somewhat weaker condition d≥3p and d,q→∞. Moreover, the two events in which a random plane curve is smooth and respectively

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 .

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.

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 .