We give explicit upper bounds for the Stirling numbers of the first kind s(n,m) which are asymptotically sharp. The form of such bounds varies according to m lying in the central or non-central regions of {1,…,n}. In each case, we use a different probabilistic representation of s(n,m) in terms of well known random variables to show the corresponding upper bounds. Some applications concerning the Riemann

Given a connected graph Δ, a group G can be constructed in such a way that Δ is often isomorphic to a subgraph of the commuting graph KG(Δ) of G. We show that, with one exception, KG(Δ) is connected, and in this latter case its diameter is at most that of Δ. If Δ is a path of length n>2, then diam(KG(Δ))=n.

The Kazhdan-Lusztig polynomial of a matroid was introduced by Elias, Proudfoot and Wakefield. The properties of these polynomials need to be further explored. In this paper we prove that the Kazhdan-Lusztig polynomials of fan matroids coincide with Motzkin polynomials, which was conjectured by Gedeon. As a byproduct, we determine the Kazhdan-Lusztig polynomials of graphic matroids of squares of paths

A well-known conjecture due to van Lint and MacWilliams states that if A is a subset of Fq2 such that 0,1∈A, |A|=q, and a−b is a square for each a,b∈A, then A must be the subfield Fq. This conjecture is often phrased in terms of the maximum cliques in Paley graphs. It was first proved by Blokhuis and later extended by Sziklai to generalized Paley graphs. In this paper, we give a new proof of the conjecture

A line packing is optimal if its coherence is as small as possible. Most interesting examples of optimal line packings are achieving equality in some of the known lower bounds for coherence. In this paper two infinite families of real and complex biangular line packings are presented. New packings achieve equality in the real or complex second Levenshtein bound respectively. Both infinite families

This work investigates the structure of rank-metric codes in connection with concepts from finite geometry, most notably the q-analogues of projective systems and blocking sets. We also illustrate how to associate a classical Hamming-metric code to a rank-metric one, in such a way that various rank-metric properties naturally translate into the homonymous Hamming-metric notions under this correspondence

In this paper, oppositeness in spherical buildings is used to define an EKR-problem for flags in projective and polar spaces. A novel application of the theory of buildings and Iwahori-Hecke algebras is developed to prove sharp upper bounds for EKR-sets of flags. In this framework, we can reprove and generalize previous upper bounds for EKR-problems in projective and polar spaces. The bounds are obtained

Abstract not available

We investigate the association schemes Inv(G) that are formed by the collection of orbitals of a permutation group G, for which the (underlying) graph Γ of a basis relation is a distance-regular antipodal cover of the complete graph. The group G can be regarded as an edge-transitive group of automorphisms of Γ and induces a 2-homogeneous permutation group on the set of its antipodal classes, which

Approximate proof labeling schemes were introduced by Censor-Hillel, Paz and Perry [3]. Roughly speaking, a graph property P can be verified by an approximate proof labeling scheme in constant-time if the vertices of a graph having the property can be convinced, in a short period of time not depending on the size of the graph, that they are having the property P or at least they are not far from being

A flag of a finite set S is a set f of non-empty proper subsets of S such that A⊆B or B⊆A for all A,B∈f. The set {|A|:A∈f} is called the type of f. Two flags f and f′ are in general position (with respect to S) when A∩B=∅ or A∪B=S for all A∈f and B∈f′. We study sets of flags of a fixed type T that are mutually not in general position and are interested in the largest cardinality of these sets. This

We study the seminormal basis {ft} for the Specht modules of the Iwahori-Hecke algebra Hn(q) of type An−1. We focus on the base change coefficients between the seminormal basis {ft} and Murphys' standard basis {xt} with emphasis on the denominators of these coefficients. In certain important cases we obtain simple formulas for these coefficients involving hook lengths. Even for general standard tableaux

We study the Pieri type formulas for the Schur multiple zeta functions along with those for the Schur functions. To express these formulas, we introduce a new insertion rule for adding boxes in the Young tableaux and obtain the results for the hook type Schur multiple zeta functions. For the proof, we show certain extended Jacobi-Trudi formulas for the Schur multiple zeta functions.

A permutation whose any prefix has no more descents than ascents is called a ballot permutation. In this paper, we present a decomposition of ballot permutations that enables us to construct a bijection between ballot permutations and odd order permutations, which proves a set-valued extension of a conjecture due to Spiro using the statistic of peak values. This bijection also preserves the neighbors

In 2003, Alladi, Andrews and Berkovich proved a four parameter partition identity lying beyond a celebrated identity of Göllnitz. Since then it has been an open problem to extend their work to five or more parameters. In part I of this pair of papers, we took a first step in this direction by giving a bijective proof of a reformulation of their result. We introduced forbidden patterns, bijectively

Groups defined by presentations for which the components of the corresponding star graph are the incidence graphs of generalized polygons are of interest as they are small cancellation groups that – via results of Edjvet and Vdovina – are fundamental groups of polyhedra with the generalized polygons as links and so act on Euclidean or hyperbolic buildings; in the hyperbolic case the groups are SQ-universal

An automorphism ρ of a graph X is said to be semiregular provided all of its cycles in its cycle decomposition are of the same length, and is said to be simplicial if it is semiregular and the quotient multigraph Xρ of X with respect to ρ is a simple graph, and thus of the same valency as X. It is shown that, with the exception of the complete graph K4, the Petersen graph, the Coxeter graph and the

Consider the Klein quadric Q+(5,q) in PG(5,q). A set of points of Q+(5,q) is called a quadratic set if it intersects each plane π of Q+(5,q) in a possibly reducible conic of π, i.e. in a singleton, a line, an irreducible conic, a pencil of two lines or the whole of π. A quadratic set is called good if at most two of these possibilities occur as π ranges over all planes of Q+(5,q). We obtain several

From the configuration of a matroid (which records the size and rank of the cyclic flats and the containments among them, but not the sets), one can compute several much-studied matroid invariants, including the Tutte polynomial and a newer, stronger invariant, the G-invariant. To gauge how much additional information the configuration contains compared to these invariants, it is of interest to have

We show how Andrews' generating functions for generalized Frobenius partitions can be understood within the theory of Eichler and Zagier as specific coefficients of certain Jacobi forms. This reformulation leads to a recursive process which yields explicit formulas for the generalized Frobenius partition generating functions in terms of infinite q-products. In particular, we show that specific examples

We obtain an explicit combinatorial formula for certain parabolic Kostka-Shoji polynomials associated with the cyclic quiver, generalizing results of Shoji and of Liu and Shoji.

The orbital diameter of a primitive permutation group is the maximal diameter of its orbital graphs. There has been a lot of interest in bounds for the orbital diameter. In this paper we provide explicit bounds on the diameters of groups of simple diagonal type. As a consequence we obtain a classification of simple diagonal groups with orbital diameter less than or equal to 4. As part of this, we classify

Let L be an n×n array whose top left r×s subarray is filled with k different symbols, each occurring at most once in each row and at most once in each column. We find necessary and sufficient conditions that ensure the remaining cells of L can be filled such that each symbol occurs at most once in each row and at most once in each column, and each symbol occurs a prescribed number of times in L. The

We present a new procedure to determine the growth function of a homogeneous Garside monoid, with respect to the finite generating set formed by the atoms. In particular, we present a formula for the growth function of each Artin–Tits monoid of spherical type (hence of each braid monoid) with respect to the standard generators, as the inverse of the determinant of a very simple matrix. Using this approach

We establish a bijection between the (n−2)-stack sortable permutations and the labeled lattice paths. Using this bijection, we directly give combinatorial proof for the log-concavity of the numbers of (n−2)-stack sortable permutations with k descents. Furthermore, we prove the numbers of (n−2)-stack sortable permutations with k descents satisfy interlacing log-concavity. We also consider a conjecture

We obtain an explicit criterion for p-ary functions to produce association schemes in terms of their Walsh spectrum. Employing this characterization, we explicitly find a correlation between p-ary bent functions and association schemes; to be more exact, we prove that a p-ary bent function induces a p-class association scheme if and only if the function is weakly regular. As applications of our main

Let I=I(G) be the edge ideal of a simple graph G. We prove thatreg(I(s))=reg(Is) for s=2,3, where I(s) is the s-th symbolic power of I. As a consequence, we prove the following boundsregIs≤regI+2s−2, for s=2,3,regI(s)≤regI+2s−2, for s=2,3,4.

This paper compares the divisorial gonality of a finite graph G to the divisorial gonality of the associated metric graph Γ(G,1) with unit lengths. We show that dgon(Γ(G,1)) is equal to the minimal divisorial gonality of all regular subdivisions of G, and we provide a class of graphs for which this number is strictly smaller than the divisorial gonality of G. This settles a conjecture of M. Baker [3

Given a graph G on the vertex set V, the non-matching complex of G, denoted by NMk(G), is the family of subgraphs G′⊂G whose matching number ν(G′) is strictly less than k. As an attempt to extend the result by Linusson, Shareshian and Welker on the homotopy types of NMk(Kn) and NMk(Kr,s) to arbitrary graphs G, we show that (i) NMk(G) is (3k−3)-Leray, and (ii) if G is bipartite, then NMk(G) is (2k−2)-Leray

In a paper published in 2005, Praeger and Zhou improved the upper bound on the number of points of a flag-transitive, point-imprimitive, symmetric design in terms of the number of blocks containing two points. Here we present an improvement of their result, devise a list of feasible parameter sequences for which two points are contained in at most ten blocks, and also develop new methods for the elimination

Let G be a multiplicatively written finite group. We denote by d(G) the small Davenport constant of G, that is, the maximal integer ℓ such that there is a sequence of length ℓ over G which has no non-trivial product-one subsequence. In 2014, Gao, Li, and Peng conjectured that d(G)≤|G|/p+p−2 for any finite non-cyclic group G, where p is the smallest prime divisor of |G|. In this paper, we confirm that

Let Fq be a finite field and G a finite abelian group. An abelian code is an ideal of Fq[G]. Two abelian codes c1 and c2 of Fq[G] are equivalent if there exists an automorphism of G whose linear extension to Fq[G] maps c1 onto c2. MacWilliams determined the number of equivalent classes of minimal abelian codes (minimal ideals) in F2[G] for cyclic group G of odd cardinality. Miller claimed that MacWilliams'

Let F be a 2-regular graph of order v. The Oberwolfach problem OP(F), posed in 1967 and still open, asks for a decomposition of Kv into copies of F. In this paper we show that OP(F) has a solution whenever F has a sufficiently large cycle which meets a given lower bound and, in addition, has a single-flip automorphism, which is an involutory automorphism acting as a reflection on exactly one of the

We show the existence of several infinite monochromatic patterns in the integers obtained as values of suitable symmetric polynomials; in particular, we obtain extensions of both the additive and multiplicative versions of Hindman's theorem. These configurations are obtained by means of suitable symmetric polynomials that mix the two operations. The simplest example is the following. For every finite

Let S denote the set of integer partitions into parts that differ by at least 3, with the added constraint that no two consecutive multiples of 3 occur as parts. We derive trivariate generating functions of Andrews–Gordon type for partitions in S with both the number of parts and the number of even parts counted. In particular, we provide an analytic counterpart of Andrews' recent refinement of the

r-fat polynomials are a natural generalization of scattered polynomials. They define linear sets of the projective line PG(1,qn) of rank n with r points of weight larger than one. Using techniques on algebraic curves and function fields, we obtain numerical bounds for r and the non-existence of exceptional r-fat polynomials with r>0. We completely determine the possible values of r when considering

We prove that the chromatic symmetric function of any tree with a vertex of degree at least six is not e-positive, that is, it cannot be written as a nonnegative linear combination of elementary symmetric functions. This makes significant progress towards a recent conjecture of Dahlberg, She, and van Willigenburg, who conjectured the result for the chromatic symmetric functions of all trees with a

We prove the non-existence of extended perfect codes in H(n,q), where q=3,4 and n>q+2 or both n and q are odd. In particular, we prove the non-existence of non-linear (q+2,qq−1,4)q MDS codes when q is odd.

A group G admits an n-partite digraphical representation if there exists a regular n-partite digraph Γ such that the automorphism group Aut(Γ) of Γ satisfies the following properties: (1) Aut(Γ) is isomorphic to G, (2) Aut(Γ) acts semiregularly on the vertices of Γ and (3) the orbits of Aut(Γ) on the vertex set of Γ form a partition into n parts giving a structure of n-partite digraph to Γ. In this

A sequence S is called r-nonrepetitive if no r sequentially adjacent blocks in S are identical. By the classic results of Thue from the beginning of the 20th century, we know that there exist arbitrarily long binary 3-nonrepetitive sequences and ternary 2-nonrepetitive sequences. This discovery stimulated over the years intensive research leading to various generalizations and many exciting problems

We show a first-order asymptotics result for the number of galled networks with n leaves. This is the first class of phylogenetic networks of large size for which an asymptotic counting result of such strength can be obtained. In addition, we also find the limiting distribution of the number of reticulation nodes of galled networks with n leaves chosen uniformly at random. These results are obtained

We prove a Touchard type identity for q-Narayana number as follows:1[n]q[nk]q[nk+1]q=∑h=0min⁡{k,n−1−k}Ch⋅qh2(n−1n−1−h−k,2h,k−h)q, where Ch=1h+1(2hh) is the original Catalan number and (nk0,k1,k2)q is a q-analogue of multinomial coefficient given by∑k0+k1+k2=n(nk0,k1,kn)q⋅q(k22)+(k1+k22)xk1yk2=∏j=0n−1(1+xqj+yq2j).

Let M denote the set Sn,q of n×n symmetric matrices with entries in Fq or the set Hn,q2 of n×n Hermitian matrices whose elements are in Fq2. Then M equipped with the rank distance dr is a metric space. We investigate d–codes in (M,dr) and construct d–codes whose sizes are larger than the corresponding additive bounds. In the Hermitian case, we show the existence of an n–code of M, n even and n/2 odd

We recover the Tutte polynomial of a matroid, up to change of coordinates, from an Ehrhart-style polynomial counting lattice points in the Minkowski sum of its base polytope and scalings of simplices. Our polynomial has coefficients of alternating sign with a combinatorial interpretation closely tied to the Dawson partition. Our definition extends in a straightforward way to polymatroids, and in this

We say a subset C⊆{1,2,…,k}n is a k-hash code (also called k-separated) if for every subset of k codewords from C, there exists a coordinate where all these codewords have distinct values. Understanding the largest possible rate (in bits), defined as (log2⁡|C|)/n, of a k-hash code is a classical problem. It arises in two equivalent contexts: (i) the smallest size possible for a perfect hash family

In this paper, we study the pattern occurrence in k-ary words. We prove an explicit upper bound on the number of k-ary words avoiding any given pattern using a random walk argument. Additionally, we reproduce one already known result on the exponential rate of growth of pattern occurrence in words and establish a simple connection among pattern occurrences in permutations and k-ary words. A simple

Squarefree powers of edge ideals are intimately related to matchings of the underlying graph. In this paper, we give bounds for the regularity of squarefree powers of edge ideals, and we consider the question of when such powers are linearly related or have linear resolution. We also consider the so-called squarefree Ratliff property.

We study incidences between points and (constant-degree algebraic) curves in three dimensions, taken from a family C of curves that have almost two degrees of freedom, meaning that (i) every pair of curves of C intersect in O(1) points, (ii) for any pair of points p, q, there are only O(1) curves of C that pass through both points, and (iii) there exists a 6-variate real polynomial F of constant degree

Let Qnr be the graph with vertex set {−1,1}n in which two vertices are joined if their Hamming distance is at most r. The edge-isoperimetric problem for Qnr is that: For every (n,r,M) such that 1≤r≤n and 1≤M≤2n, determine the minimum edge-boundary size of a subset of vertices of Qnr with a given size M. In this paper, we apply two different approaches to prove bounds for this problem. The first approach

Recently a new generalization of Pascal's triangle, the family of so-called hyperbolic Pascal triangles (in short HPT) was introduced. In this paper, we consider a variation of HPT by replacing the two leg-sequences of the triangle by arbitrary sequences {αn}n≥0 and {βn}n≥0. Originally the legs were the constant 1 sequence. We describe some quantitative properties via recurrences relations, explicit

The multicolor Ramsey number problem asks, for each pair of natural numbers ℓ and t, for the largest ℓ-coloring of a complete graph with no monochromatic clique of size t. Recent works of Conlon-Ferber and Wigderson have improved the longstanding lower bound for this problem. We make a further improvement by replacing an explicit graph appearing in their constructions by a random graph. Graphs useful

In this paper, we derive some restrictions and nonexistence results for (v,m,k,pq)-strong external difference families (SEDFs), where p and q are primes. We first show that there is no abelian (v,m,k,p2)-SEDF with m>2. If p>q, we show that if q+1 is a power of two; or q+1=2r or 4r for some prime r>3, then there is no abelian (v,m,k,pq)-SEDF with m>2 for all sufficiently large primes p. Furthermore

This paper completely characterizes the standard Young tableaux that can be reconstructed from their sets or multisets of 1-minors. In particular, any standard Young tableau with at least 5 entries can be reconstructed from its set of 1-minors.

Given an element x of a commutative quasigroup A, the x-divisor graph of A is the graph Γx(A) whose vertices are the elements of A such that distinct vertices a and b are adjacent if and only if ab=x. This paper examines how isotopies act on the set {Γx(A) | x∈A}. It is shown that if |A|<∞ is not a multiple of 4, then this set is an isotopy invariant. Moreover, under certain conditions (for example

The spectral properties of signed directed graphs, which may be naturally obtained by assigning a sign to each edge of a directed graph, have received substantially less attention than those of their undirected and/or unsigned counterparts. To represent such signed directed graphs, we use a striking equivalence to T6-gain graphs to formulate a Hermitian adjacency matrix, whose entries are the unit

For a graph G, its Tutte symmetric function XBG generalizes both the Tutte polynomial TG and the chromatic symmetric function XG. We may also consider XB as a map from the t-extended Hopf algebra G[t] of labeled graphs to symmetric functions. We show that the kernel of XB is generated by vertex-relabellings and a finite set of modular relations, in the same style as a recent analogous result by Penaguião

We propose a discrete approach to solve problems on forming polygons from broken sticks, which is akin to counting polygons with sides of integer length subject to certain Diophantine inequalities. Namely, we use MacMahon's Partition Analysis to obtain a generating function for the size of the set of segments of a broken stick subject to these inequalities. In particular, we use this approach to show

A binary Steinhaus triangle is a triangle of zeroes and ones that points down and with the same local rule as the Pascal triangle modulo 2. A binary Steinhaus triangle is said to be rotationally symmetric, horizontally symmetric or dihedrally symmetric if it is invariant under the 120 degrees rotation, the horizontal reflection or both, respectively. The first part of this paper is devoted to the study

Let G be an additive finite abelian group of exponent exp⁡(G). For every positive integer k, let skexp⁡(G)(G) denote the smallest integer t such that every sequence over G of length t has a zero-sum subsequence of length kexp⁡(G). Let ηkexp⁡(G)(G) denote the smallest integer t such that every sequence over G of length t has a zero-sum subsequence of length between 1 and kexp⁡(G). It is conjectured

One of the most classical results in extremal set theory is Sperner's theorem, which says that the largest antichain in the Boolean lattice 2[n] has size Θ(2nn). Motivated by an old problem of Erdős on the growth of infinite Sidon sequences, in this note we study the growth rate of maximum infinite antichains. Using the well known Kraft's inequality for prefix codes, it is not difficult to show that