Let H be a hypergraph on a finite set V. A cover of H is a set of vertices that meets all edges of H. If W is not a cover of H, then W is said to be a noncover of H. The noncover complex of H is the abstract simplicial complex whose faces are the noncovers of H. In this paper, we study homological properties of noncover complexes of hypergraphs. In particular, we obtain an upper bound on their Leray

Within the framework of unitary easy quantum groups, we study an analogue of Brauer's Schur-Weyl approach to the representation theory of the orthogonal group. We consider concrete combinatorial categories whose morphisms are formed by partitions of finite sets into disjoint subsets of cardinality two; the points of these sets are colored black or white. These categories correspond to “half-liberated

Let A be a set of integers which is dense in a finite interval. We establish upper and lower bounds for the longest regularly-spaced and convex sequences in A and in A−A.

Amongst d-regular r-uniform hypergraphs on n vertices, which ones have the largest number of independent sets? While the analogous problem for graphs (originally raised by Granville) is now well-understood, it is not even clear what the correct general conjecture ought to be; our goal here is to propose such a generalisation. Lending credence to our conjecture, we verify it within the class of ‘quasi-bipartite’

A bisection of a graph is a bipartition of its vertex set in which the number of vertices in the two parts differ by at most 1, and its size is the number of edges which go across the two parts. In this paper, motivated by several known results of Alon, Bollobás, Krivelevich and Sudakov about Max-Cut, we study maximum bisections of graphs without short even cycles. Let G be a graph on m edges without

In this article we show how to compute the chromatic quasisymmetric function of indifference graphs from the modular law introduced in [19]. We provide an algorithm which works for any function that satisfies this law, such as unicellular LLT polynomials. When the indifference graph has bipartite complement it reduces to a planar network, in this case, we prove that the coefficients of the chromatic

A partial transversal T of a Latin square L is a set of entries of L in which each row, column and symbol is represented at most once. A partial transversal is maximal if it is not contained in a larger partial transversal. Any maximal partial transversal of a Latin square of order n has size at least ⌈n2⌉ and at most n. We say that a Latin square is omniversal if it possesses a maximal partial transversal

We investigate small weight codewords of the p-ary linear code Cj,k(n,q) generated by the incidence matrix of k-spaces and j-spaces of PG(n,q) and its dual, with q a prime power and 0⩽j

We prove that, given a finite graph Σ satisfying some mild conditions, there exist infinitely many tetravalent half-arc-transitive normal covers of Σ. Applying this result, we establish the existence of infinite families of finite tetravalent half-arc-transitive graphs with certain vertex stabilizers, and classify the vertex stabilizers up to order 28 of finite connected tetravalent half-arc-transitive

We construct a family of Sn modules indexed by c∈{1,…,n} with the property that upon restriction to Sn−1 they recover the classical parking function representation of Haiman. The construction of these modules relies on an Sn-action on a set that is closely related to the set of parking functions. We compute the characters of these modules and use the resulting description to classify them up to isomorphism

For an odd integer r>0 and an integer n>r, we introduce a notion of weakly r-separated collections of subsets of [n]={1,2,…,n}. When r=1, this corresponds to the concept of weak separation introduced by Leclerc and Zelevinsky. In this paper, extending results due to Leclerc-Zelevinsky, we develop a geometric approach to establish a number of nice combinatorial properties of maximal weakly r-separated

Given a permutation group G, the derangement graph ΓG of G is the Cayley graph with connection set the set of all derangements of G. We prove that, when G is transitive of degree at least 3, ΓG contains a triangle. The motivation for this work is the question of how large can be the ratio of the independence number of ΓG to the size of the stabilizer of a point in G. We give examples of transitive

The Dushnik-Miller dimension of a partially-ordered set P is the smallest d such that one can embed P into a product of d linear orders. We prove that the dimension of the divisibility order on the interval {1,…,n}, is equal to (log⁡n)2(log⁡log⁡n)−Θ(1) as n goes to infinity. We prove similar bounds for the 2-dimension of divisibility in {1,…,n}, where the 2-dimension of a poset P is the smallest d

We study the Hn(0)-module Sασ due to Tewari and van Willigenburg, which was constructed using new combinatorial objects called standard permuted composition tableaux and decomposed into cyclic submodules. First, we show that every direct summand appearing in their decomposition is indecomposable and characterize when Sασ is indecomposable. Second, we find characteristic relations among Sασ's and expand

A sequence e=e1e2⋯en of natural numbers is called an inversion sequence if 0≤ei≤i−1 for all i∈{1,2,…,n}. Recently, Martinez and Savage initiated an investigation of inversion sequences that avoid patterns of relation triples. Let ρ1, ρ2 and ρ3 be among the binary relations {<,>,≤,≥,=,≠,−}. Martinez and Savage defined In(ρ1,ρ2,ρ3) as the set of inversion sequences of length n such that there are no

For two elements v and w of the symmetric group Sn with v≤w in Bruhat order, the Bruhat interval polytope Qv,w is the convex hull of the points (z(1),…,z(n))∈Rn with v≤z≤w. It is known that the Bruhat interval polytope Qv,w is the moment map image of the Richardson variety Xw−1v−1. We say that Qv,w is toric if the corresponding Richardson variety Xw−1v−1 is a toric variety. We show that when Qv,w is

Veit Elser proposed a random graph model for percolation in which physical dimension appears as a parameter. Studying this model combinatorially leads naturally to the consideration of numerical graph invariants which we call Elser numbers elsk(G), where G is a connected graph and k a nonnegative integer. Elser had proven that els1(G)=0 for all G. By interpreting the Elser numbers as reduced Euler

Let f(X)∈Fqr[X] be a q-polynomial. If the Fq-subspace U={(xqt,f(x))|x∈Fqn} defines a maximum scattered linear set, then we call f(X) a scattered polynomial of index t. The asymptotic behavior of scattered polynomials of index t is an interesting open problem. In this sense, exceptional scattered polynomials of index t are those for which U is a maximum scattered linear set in PG(1,qmr) for infinitely

We study the problem of finding zero-sum blocks in bounded-sum sequences, which was introduced by Caro, Hansberg, and Montejano. Caro et al. determine the minimum {−1,1}-sequence length for when there exist k consecutive terms that sum to zero. We determine the corresponding minimum sequence length when the set {−1,1} is replaced by {−r,s} for arbitrary positive integers r and s. This confirms a conjecture

We consider solutions of the word equation X12⋯Xn2=(X1⋯Xn)2 such that the squares Xi2 are minimal squares found in optimal squareful infinite words. We apply a method developed by the second author for studying word equations and prove that there are exactly two families of solutions: reversed standard words and words obtained from reversed standard words by a simple substitution scheme. A particular

We examine the asymptotics of a class of banded plane partitions under a varying bandwidth parameter m, and clarify the transitional behavior for large size n and increasing m=m(n) to be from c1n−1exp⁡(c2n1/2) to c3n−49/72exp⁡(c4n2/3+c5n1/3) for some explicit coefficients c1,…,c5. The method of proof, which is a unified saddle-point analysis for all phases, is general and can be extended to other classes

In this paper, we completely determine the point regular automorphism groups of the Payne derived quadrangle of the symplectic quadrangle W(q), q odd. As a corollary, we show that the finite groups that act regularly on the points of a finite generalized quadrangle can have unbounded nilpotency class.

A well-known conjecture, often attributed to Ryser, states that the cover number of an r-partite r-uniform hypergraph is at most r−1 times larger than its matching number. Despite considerable effort, particularly in the intersecting case, this conjecture remains wide open, motivating the pursuit of variants of the original conjecture. Recently, Bustamante and Stein and, independently, Király and Tóthmérész

We relate the character theory of the symmetric groups S2n and S2n+1 with that of the hyperoctahedral group Bn=(Z/2)n⋊Sn, as part of the expectation that the character theory of reductive groups with diagram automorphism and their Weyl groups, is related to the character theory of the fixed subgroup of the diagram automorphism.

We develop an analytical method to prove congruences of the type∑k=0(pr−1)/dAkzk≡ω(z)∑k=0(pr−1−1)/dAkzpk(modpmrZp[[z]])forr=1,2,…, for primes p>2 and fixed integers m,d⩾1, where f(z)=∑k=0∞Akzk is an ‘arithmetic’ hypergeometric series. Such congruences for m=d=1 were introduced by Dwork in 1969 as a tool for p-adic analytical continuation of f(z). Our proofs of several Dwork-type congruences corresponding

For an integer d≥2, a family F of sets is d-wise intersecting if for any distinct sets A1,A2,…,Ad∈F, A1∩A2∩…∩Ad≠∅, and non-trivial if ⋂A∈FA=∅. Hilton and Milner conjectured that for k≥d≥2 and large enough n, the extremal (i.e. largest) non-trivial d-wise intersecting family of k-element subsets of [n] is, up to isomorphism, one of the following two families:A(k,d)={A∈([n]k):|A∩[d+1]|≥d}H(k,d)={A∈([n]k):[d−1]⊂A

We study certain bijection between plane partitions and N-matrices. As applications, we prove a Cauchy-type identity for generalized dual Grothendieck polynomials. We introduce two statistics on plane partitions, whose generating functions are similar to classical MacMahon's formulas; one of these statistics is equidistributed with the usual volume. We also show natural connections with the longest

We consider the Ehrhart polynomial of hypersimplices. It is proved that these polynomials have positive coefficients and we give a combinatorial formula for each of them. This settles a problem posed by Stanley and also proves that uniform matroids are Ehrhart positive, an important and yet unsolved particular case of a conjecture posed by De Loera et al. To this end, we introduce a new family of numbers

We prove the following Helly-type result. Let C1,…,C3d be finite families of convex bodies in Rd. Assume that for any colorful selection of 2d sets, Cik∈Cik for each 1≤k≤2d with 1≤i1<…

In this paper we consider arbitrary hexagons on the triangular lattice with three arbitrary bowtie-shaped holes, whose centers form an equilateral triangle. The number of lozenge tilings of such general regions is not expected — and indeed is not — given by a simple product formula. However, when considering a certain natural normalized counterpart R of any such region R, we prove that the ratio between

For numerical semigroups with a specified list of (not necessarily minimal) generators, we obtain explicit asymptotic expressions, and in some cases quasipolynomial/quasirational representations, for all major factorization length statistics. This involves a variety of tools that are not standard in the subject, such as algebraic combinatorics (Schur polynomials), probability theory (weak convergence

Each point x in Gr(r,n) corresponds to an r×n matrix Ax which gives rise to a matroid Mx on its columns. Gel'fand, Goresky, MacPherson, and Serganova showed that the sets {y∈Gr(r,n)|My=Mx} form a stratification of Gr(r,n) with many beautiful properties. However, results of Mnëv and Sturmfels show that these strata can be quite complicated, and in particular may have arbitrary singularities. We study

Andrews and Sellers recently initiated the study of arithmetic properties of Fishburn numbers. In this paper we prove prime power congruences for generalized Fishburn numbers. These numbers are the coefficients in the 1−q expansion of the Kontsevich-Zagier series Ft(q) for the torus knots T(3,2t), t≥2. The proof uses a strong divisibility result of Ahlgren, Kim and Lovejoy and a new “strange identity”

Type A Demazure atoms are pieces of Schur functions, or sets of tableaux whose weights sum to such functions. Inspired by colored vertex models of Borodin and Wheeler, we will construct solvable lattice models whose partition functions are Demazure atoms; the proof of this makes use of a Yang-Baxter equation for a colored five-vertex model. As a byproduct, we will construct Demazure atoms on Kashiwara's

We compute the Chow ring of an arbitrary heavy/light Hassett space M‾0,w. These spaces are moduli spaces of weighted pointed stable rational curves, where the associated weight vector w consists of only heavy and light weights. Work of Cavalieri et al. [3] exhibits these spaces as tropical compactifications of hyperplane arrangement complements. The computation of the Chow ring then reduces to intersection

Over a field K of characteristic p, let Z be the incidence variety in Pd×(Pd)⁎ and let L be the restriction to Z of the line bundle O(−n−d)⊠O(n), where n=p+f with 0≤f≤p−2. We prove that Hd(Z,L) is the simple GLd+1-module corresponding to the partition λf=(p−1+f,p−1,f+1). When f=0, using the first author's description of Hd(Z,L) and Jantzen's sum formula, we obtain as a by-product that the sum of the

In this paper we prove that a uniformly distributed random circular automaton An of order n synchronizes with high probability (w.h.p.). More precisely, we prove thatP[An synchronizes]=1−O(1n). The main idea of the proof is to translate the synchronization problem into a problem concerning properties of a random matrix; these properties are then established with high probability by a careful analysis

The notion of weak saturation was introduced by Bollobás in 1968. Let F and H be graphs. A spanning subgraph G⊆F is weakly (F,H)-saturated if it contains no copy of H but there exists an ordering e1,…,et of E(F)∖E(G) such that for each i∈[t], the graph G∪{e1,…,ei} contains a copy H′ of H such that ei∈H′. Define wsat(F,H) to be the minimum number of edges in a weakly (F,H)-saturated graph. In this paper

Let Kn be the set of all nonsingular n×n lower triangular (0,1)-matrices. Hong and Loewy (2004) introduced the numberscn=min{λ|λis an eigenvalue ofXXT,X∈Kn},n∈Z+. A related family of numbers was considered by Ilmonen, Haukkanen, and Merikoski (2008):Cn=max{λ|λis an eigenvalue ofXXT,X∈Kn},n∈Z+. These numbers can be used to bound the singular values of matrices belonging to Kn and they appear, e.g.,

An s-geodesic of a graph is a path of length s such that the first and last vertices are at distance s. We study finite graphs Γ of diameter at least 3 for which some subgroup G of automorphisms is transitive on the set of s-geodesics for each s≤3. If Γ has girth at least 6 then all 3-arcs are 3-geodesics so Γ is 3-arc-transitive, and such graphs have already been studied fruitfully; also graphs of

We prove refined enumeration results on several symmetry classes as well as related classes of alternating sign matrices with respect to classical boundary statistics, using the six-vertex model of statistical physics. More precisely, we study vertically symmetric, vertically and horizontally symmetric, vertically and horizontally perverse, off-diagonally and off-antidiagonally symmetric, vertically

This paper concerns the study of a class of clutters called simplicial subclutters. Given a clutter C and its simplicial subclutter D, we compare some algebraic properties and invariants of the ideals I,J associated to these two clutters, respectively. We give a formula for computing the (multi)graded Betti numbers of J in terms of those of I and some combinatorial data about D. As a result, we see

For an odd integer n=2d−1, let B(n,d) be the subgraph of the hypercube Qn induced by the two largest layers. In this paper, we describe the typical structure of independent sets in B(n,d) and give precise asymptotics on the number of them. The proofs use Sapozhenko's graph container method and a recently developed method of Jenssen and Perkins, which combines Sapozhenko's graph container lemma with

Let v be a product of at most three not necessarily distinct primes. We prove that there exists no strong external difference family with more than two subsets in an abelian group G of order v, except possibly when G=Cp3 and p is a prime greater than 3×1012.

In 1971, Ruzsa conjectured that if f:N→Z with f(n+k)≡f(n) mod k for every n,k∈N and f(n)=O(θn) with θ

Halved monotone triangles are a generalisation of vertically symmetric alternating sign matrices (VSASMs). We provide a weighted enumeration of halved monotone triangles with respect to a parameter which generalises the number of −1s in a VSASM. Among other things, this enables us to establish a generating function for vertically symmetric alternating sign trapezoids. Our results are mainly presented

A set of integers is sum-free if it does not contain any solution for x+y=z. Answering a question of Cameron and Erdős, Balogh, Liu, Sharifzadeh and Treglown recently proved that the number of maximal sum-free sets in {1,…,n} is Θ(2μ(n)/2), where μ(n) is the size of a largest sum-free set in {1,…,n}. They conjectured that, in contrast to the integer setting, there are abelian groups G having exponentially

The slice-rank method, introduced by Tao as a symmetrized version of the polynomial method of Croot, Lev and Pach and Ellenberg and Gijswijt, has proved to be a useful tool in a variety of combinatorial problems. Explicit tensors have been introduced in different contexts but little is known about the limitations of the method. In this paper, building upon a method presented by Tao and Sawin, it is

We prove the equivalence of EL-shellability and the existence of recursive atom ordering independent of roots. We show that a comodernistic lattice, as defined by Schweig and Woodroofe, admits a recursive atom ordering independent of roots, therefore is EL-shellable. We also present and discuss a simpler EL-shelling on one of the most important classes of comodernistic lattice, the order congruence

Since its formulation, Turán's hypergraph problems have been among the most challenging open problems in extremal combinatorics. One of them is the following: given a 3-uniform hypergraph F on n vertices in which any five vertices span at least one edge, prove that |F|≥(1/4−o(1))(n3). The construction showing that this bound would be best possible is simply (X3)∪(Y3) where X and Y evenly partition

In 2011, Penttila and Williford constructed an infinite new family of primitive Q-polynomial 3-class association schemes, not arising from distance regular graphs, by exploring the geometry of the lines of the unitary polar space H(3,q2), q even, with respect to a symplectic polar space W(3,q) embedded in it. In a private communication to Penttila and Williford, H. Tanaka pointed out that these schemes

Creation operators act on symmetric functions to build Schur functions, Hall–Littlewood polynomials, and related symmetric functions one row at a time. Haglund, Morse, Zabrocki, and others have studied more general symmetric functions Hα, Cα, and Bα obtained by applying any sequence of creation operators to 1. We develop new combinatorial models for the Schur expansions of these and related symmetric

We provide general closed-form formulas for the index of type-A Lie poset algebras corresponding to posets of restricted height. Furthermore, we provide a combinatorial recipe for constructing all posets corresponding to type-A Frobenius Lie poset algebras of heights zero, one, and two. A discrete Morse theory argument establishes that the simplicial realizations of such posets are contractible. It

A graph G is said to be chordal if it has no induced cycles of length four or more. In a recent preprint Culbertson, Guralnik, and Stiller give a new characterization of chordal graphs in terms of sequences of what they call ‘edge-erasures’. We show that these moves are in fact equivalent to a linear quotient ordering on IG‾, the edge ideal of the complement graph. Known results imply that IG‾ has

Let Γ(x0) be a finite rooted tree, for which Γ is the underlying tree and x0 the root. Let T be the Terwilliger algebra of Γ with respect to x0. We study the structure of the principal T-module. As a result, it is shown that T recognizes the isomorphism class of Γ(x0).

In the study of aperiodic order and mathematical models of quasicrystals, questions regarding equivalence relations on Delone sets naturally arise. This work is dedicated to the bounded displacement (BD) equivalence relation, and especially to results concerning instances of non-equivalence. We present a general condition for two Delone sets to be BD non-equivalent, and apply our result to Delone sets

Let Γ be a Q-polynomial distance-regular graph of diameter d≥3. Fix a vertex γ of Γ and consider the subgraph induced on the union of the last two subconstituents of Γ with respect to γ. We prove that this subgraph is connected.

This paper presents a combinatorial study of sums of integer powers of the cotangent. Our main tool is the realization of the cotangent values as eigenvalues of a simple self-adjoint matrix with complex integer entries. We use the trace method to draw conclusions about integer values of the sums and series expansions of the generating function to provide explicit evaluations; it is remarkable that

A compacted binary tree is a directed acyclic graph encoding a binary tree in which common subtrees are factored and shared, such that they are represented only once. We show that the number of compacted binary trees of size n grows asymptotically likeΘ(n!4ne3a1n1/3n3/4), where a1≈−2.338 is the largest root of the Airy function. Our method involves a new two parameter recurrence which yields an algorithm

Let (G,1G) be a finite group and let S=g1⋅…⋅gℓ be a nonempty sequence over G. We say S is a tiny product-one sequence if its terms can be ordered such that their product equals 1G and ∑i=1ℓ1ord(gi)≤1. Let ti(G) be the smallest integer t such that every sequence S over G with |S|≥t has a tiny product-one subsequence. The direct problem is to obtain the exact value of ti(G), while the inverse problem