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

We define non-commutative versions of the vertex packing polytope, the theta convex body and the fractional vertex packing polytope of a graph, and establish a quantum version of the Sandwich Theorem of Grötschel, Lovász and Schrijver (1986) [7]. We define new non-commutative versions of the Lovász number of a graph which lead to an upper bound of the zero-error capacity of the corresponding quantum

This paper is devoted to the classification of flag-transitive 2-(v,k,2) designs. We show that apart from two known symmetric 2-(16,6,2) designs, every flag-transitive subgroup G of the automorphism group of a nontrivial 2-(v,k,2) design is primitive of affine or almost simple type. Moreover, we classify the 2-(v,k,2) designs admitting a flag transitive almost simple group G with socle PSL(n,q) for

The aim of this work is to study the v-number of edge ideals of clutters. We relate the v-number with the regularity of edge ideals and classify W2 graphs. If the edge ideal of a graph has second symbolic power Cohen–Macaulay, we show that the graph is edge-critical.

A Leonard pair is an ordered pair of diagonalizable linear maps on a finite-dimensional vector space, that each act on an eigenbasis for the other one in an irreducible tridiagonal fashion. In the present paper we consider a type of Leonard pair, said to have spin. The notion of a spin model was introduced by V.F.R. Jones to construct link invariants. A spin model is a symmetric matrix over C that

We prove that every 2-transitive group has a property called the EKR-module property. This property gives a characterization of the maximum intersecting sets of permutations in the group. Specifically, the characteristic vector of any maximum intersecting set in a 2-transitive group is a linear combination of the characteristic vectors of the stabilizers of points and their cosets. We also consider

It is known that unicellular LLT polynomials are related to the quasi-symmetric chromatic polynomials of certain graphs by the (t−1)-transform of symmetric functions. We investigate the extension of this transformation to various combinatorial Hopf algebras and prove a noncommutative version of this property.

Schubitopes were introduced by Monical, Tokcan and Yong as a specific family of generalized permutohedra. It was proven by Fink, Mészáros and St. Dizier that Schubitopes are the Newton polytopes of the dual characters of flagged Weyl modules. Important cases of Schubitopes include the Newton polytopes of Schubert polynomials and key polynomials. In this paper, we develop a combinatorial rule to generate

For a polynomial ring S in n variables, we consider the natural action of the symmetric group Sn on S by permuting the variables. For an Sn-invariant monomial ideal I⊆S and j≥0, we give an explicit recipe for computing the modules ExtSj(S/I,S), and use this to describe the projective dimension and regularity of I. We classify the Sn-invariant monomial ideals I that have a linear free resolution, and

We consider a discrete non-deterministic flow-firing process for rerouting flow on the edges of a planar complex. The process is an instance of higher-dimensional chip-firing. In the flow-firing process, flow on the edges of a complex is repeatedly diverted across the faces of the complex. For non-conservative initial configurations we show this process never terminates. For conservative initial flows

We introduce the notion of a Whitney dual of a graded poset. Two posets are Whitney duals to each other if (the absolute value of) their Whitney numbers of the first and second kind are interchanged between the two posets. We define new types of edge labelings which we call Whitney labelings. We prove that every graded poset with a Whitney labeling has a Whitney dual. Moreover, we show how to explicitly

We prove that the number of 01-fillings of a given stack polyomino (a polyomino with justified rows whose lengths form a unimodal sequence) with at most one 1 per column which do not contain a fixed-size northeast chain and a fixed-size southeast chain, depends only on the set of row lengths of the polyomino. The proof is via a bijection between fillings of stack polyominoes which differ only in the

An ordered r-graph is an r-uniform hypergraph whose vertex set is linearly ordered. Given 2≤k≤r, an ordered r-graph H is interval k-partite if there exist at least k disjoint intervals in the ordering such that every edge of H has nonempty intersection with each of the intervals and is contained in their union. Our main result implies that if α>k−1, then for each d>0 every n-vertex ordered r-graph

We generalise the lattice word condition from Young tableaux to all Kronecker tableaux and hence calculate a large new family of stable Kronecker coefficients.

In 2011, Fang et al. in [9] posed the following problem: Classify non-normal locally primitive Cayley graphs of finite simple groups of valency d, where either d≤20 or d is a prime number. The only case for which the complete solution of this problem is known is of d=3. Except this, a lot of efforts have been made to attack this problem by considering the following problem: Characterize finite nonabelian

Motivated by the study of simultaneous cores, we give three proofs (in varying levels of generality) that the expected norm of a weight in a highest weight representation Vλ of a complex simple Lie algebra g is 1h+1〈λ,λ+2ρ〉. First, we argue directly using the polynomial method and the Weyl character formula. Second, we relate this problem to the “Winnie-the-Pooh problem” regarding orthogonal decompositions

Let t be an integer such that t≥2. Let K2,t(3) denote the triple system consisting of the 2t triples {a,xi,yi}, {b,xi,yi} for 1≤i≤t, where the elements a,b,x1,x2,…,xt, y1,y2,…,yt are all distinct. Let ex(n,K2,t(3)) denote the maximum size of a triple system on n elements that does not contain K2,t(3). This function was studied by Mubayi and Verstraëte [9], where the special case t=2 was a problem of

Castelnuovo-Mumford regularity is a measure of algebraic complexity of an ideal. Regularity of monomial ideals can be investigated combinatorially. We use a simple graph decomposition and results from structural graph theory to prove, improve and generalize many of the known bounds on regularity of quadratic square-free monomial ideals.

We study involutive set-theoretic solutions of the Yang-Baxter equation of multipermutation level 2. These solutions happen to fall into two classes – distributive ones and non-distributive ones. The distributive ones can be effectively constructed using a set of abelian groups and a matrix of constants. Using this construction, we enumerate all distributive involutive solutions up to size 14. The

A new probabilistic technique for establishing the existence of certain regular combinatorial structures has been introduced by Kuperberg, Lovett, and Peled (STOC 2012). Using this technique, it can be shown that under certain conditions, a randomly chosen structure has the required properties of a t-(n,k,λ) combinatorial design with tiny, yet positive, probability. The proof method of KLP is adapted

A syntax tree is a planar rooted tree where internal nodes are labeled on a graded set of generators. There is a natural notion of occurrence of contiguous pattern in such trees. We describe a way, given a set of generators G and a set of patterns P, to enumerate the trees constructed on G and avoiding P. The method is built around inclusion-exclusion formulas forming a system of equations on formal

We propose a combinatorial approach to the following strengthening of Gal's conjecture: γ(Δ)≥γ(E) coefficientwise, where Δ is a flag homology sphere and E⊆Δ an induced homology sphere of codimension 1. We provide partial evidence in favor of this approach, and prove a nontrivial nonlinear inequality that follows from the above conjecture, for boundary complexes of flag d-polytopes: h1(Δ)hi(Δ)≥(d−i

Two words are k-binomially equivalent whenever they share the same subwords, i.e., subsequences, of length at most k with the same multiplicities. This is a refinement of both the abelian equivalence and the Simon congruence. The k-binomial complexity of an infinite word x maps the integer n to the number of classes in the quotient, by this k-binomial equivalence relation, of the set of factors of

In this article we study mutation of friezes of type D. We provide a combinatorial formula for the entries in a frieze after mutation. The two main ingredients in the proof include a certain transformation of a type D frieze into a sub-pattern of a frieze of type A and the mutation formula for type A friezes recently found by Baur et al.

We construct a family of hemisystems of the parabolic quadric Q(2d,q), for all ranks d⩾2 and all odd prime powers q, that admit Ω3(q)≅PSL2(q). This yields the first known construction for d⩾4.

Linial and Morgenstern conjectured that, among all n-vertex tournaments with d(n3) cycles of length three, the number of cycles of length four is asymptotically minimized by a random blow-up of a transitive tournament with all but one part of equal size and one smaller part. We prove the conjecture for d≥1/36 by analyzing the possible spectrum of adjacency matrices of tournaments. We also demonstrate

Let n be a positive integer and I a k-subset of integers in [0,n−1]. Given a k-tuple A=(α0,⋯,αk−1)∈Fqnk, let MA,I denote the matrix (αiqj) with 0≤i≤k−1 and j∈I. When I={0,1,⋯,k−1}, MA,I is called a Moore matrix which was introduced by E. H. Moore in 1896. It is well known that the determinant of a Moore matrix equals 0 if and only if α0,⋯,αk−1 are Fq-linearly dependent. We call I that satisfies this

In this paper, we develop a new method for constructing m-ovoids in the symplectic polar space W(2r−1,pe) from some strongly regular Cayley graphs constructed in [6]. Using this method, we obtain many new m-ovoids which can not be derived by field reduction.

We present unified bijections for maps on the torus with control on the face-degrees and essential girth (girth of the periodic planar representation). A first step is to show that for d≥3 every toroidal d-angulation of essential girth d can be endowed with a certain ‘canonical’ orientation (formulated as a weight-assignment on the half-edges). Using an adaptation of a construction by Bernardi and

We introduce the notion of a quasi-matroidal class of ordered simplicial complexes: an approximation to the idea of a matroid cryptomorphism in the landscape of ordered simplicial complexes. A quasi-matroidal class contains pure shifted simplicial complexes and ordered matroid independence complexes. The essential property is that if a fixed simplicial complex belongs to this class for every ordering

The present article exploits the fact that permutads (aka shuffle algebras) are algebras over a terminal operad in a certain operadic category Per. In the first, classical part we formulate and prove a claim envisaged by Loday and Ronco that the cellular chains of the permutohedra form the minimal model of the terminal permutad which is moreover, in the sense we define, self-dual and Koszul. In the

In the 1995 paper entitled “Noncommutative symmetric functions”, Gelfand et al. defined two noncommutative symmetric function analogues for the power sum basis of the symmetric functions. This paper explores the combinatorial properties of their duals, two distinct quasisymmetric power sum bases. In contrast to the symmetric power sums, the quasisymmetric power sums have a more complex combinatorial

We exhibit a bijection between recently-introduced combinatorial objects known as valid hook configurations and certain weighted set partitions. When restricting our attention to set partitions that are matchings, we obtain three new combinatorial interpretations of Lassalle's sequence. One of these interpretations involves permutations that have exactly one preimage under the (West) stack-sorting

We give a complete characterization of graphs whose binomial edge ideal is licci. An important tool is a new general upper bound for the regularity of binomial edge ideals.

Gorsky and Negut introduced operators Qm,n on symmetric functions and conjectured that, in the case where m and n are relatively prime, the expression Qm,n(1) is given by the Hikita polynomial Hm,n[X;q,t]. Later, Bergeron-Garsia-Leven-Xin extended and refined the conjectures of Qm,n(1) for arbitrary m and n which we call the Extended Rational Shuffle Conjecture. In the special case Qn+1,n(1), the Rational

The Shuffle Theorem of Carlsson and Mellit gives a combinatorial expression for the bigraded Frobenius characteristic of the ring of diagonal harmonics, and the Delta Conjecture of Haglund, Remmel and the second author provides two generalizations of the Shuffle Theorem to the delta operator expression Δek′en. Haglund et al. also propose the Extended Delta Conjecture for the delta operator expression

The chromatic symmetric function on graphs is a celebrated graph invariant. Analogous chromatic maps can be defined on other objects, as presented by Aguiar, Bergeron and Sottile. The problem of identifying the kernel of some of these maps was addressed by Féray, for the Gessel quasisymmetric function on posets. On graphs, we show that the modular relations and isomorphism relations span the kernel

A self-avoiding walk (SAW) is a path on a graph that visits each vertex at most once. The mean square displacement of an n-step SAW is the expected value of the square of the distance between its ending point and starting point, where the expectation is taken with respect to the uniform measure on n-step SAWs starting from a fixed vertex. It is conjectured that the mean square displacement of an n-step

In this note we investigate correlation inequalities for ‘up-sets’ of permutations, in the spirit of the Harris–Kleitman inequality. We focus on two well-studied partial orders on Sn, giving rise to differing notions of up-sets. Our first result shows that, under the strong Bruhat order on Sn, up-sets are positively correlated (in the Harris–Kleitman sense). Thus, for example, for a (uniformly) random

We prove that among all flag homology 5-manifolds with n vertices, the join of 3 circles of as equal length as possible is the unique maximizer of all the face numbers. The same upper bounds on the face numbers hold for 5-dimensional flag Eulerian normal pseudomanifolds.

We show that in every 3-coloring of the edges of a graph G of order N such that δ(G)≥5N6−1, there is a monochromatic component of order at least N/2. We also show that this result is best possible.

We introduce a simple method for proving lower bounds for the size of the smallest percolating set in a certain graph bootstrap process. We apply this method to obtain recursive formulas for the sizes of the smallest percolating sets in multidimensional tori and multidimensional grids (in particular hypercubes). The former answers a question of Morrison and Noel [9], and the latter provides an alternative

In this paper, we study a simplicial complex on the elements of the Tamari lattice in types A and B called the canonical join complex. The canonical join representation of an element w in a lattice L is the unique lowest expression ⋁A for w, when such an expression exists. We say that the set A is a canonical join representation. The collection of all such subsets has the structure of an abstract simplicial

A k-positive matrix is a matrix where all minors of order k or less are positive. Computing all such minors to test for k-positivity is inefficient, as there are ∑ℓ=1k(nℓ)2 of them in an n×n matrix. However, there are minimal k-positivity tests which only require testing n2 minors. These minimal tests can be related by series of exchanges, and form a family of sub-cluster algebras of the cluster algebra

Let G be a finite abelian group, and r be a multiple of its exponent. The generalized Erdős–Ginzburg–Ziv constant sr(G) is the smallest integer s such that every sequence of length s over G has a zero-sum subsequence of length r. We find exact values of s2m(Z2d) for d≤2m+1. Connections to linear binary codes of maximal length and codes without a forbidden weight are discussed.

We use Galois theory of difference equations to study the nature of the generating series of (weighted) walks in the quarter plane with genus zero kernel curve. Using this approach, we prove that the generating series do not satisfy any nontrivial (possibly nonlinear) algebraic differential equation with rational coefficients.