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.

Hurwitz numbers enumerate ramified coverings of the Riemann sphere with fixed ramification data. Certain kinds of ramification data are of particular interest, such as double Hurwitz numbers, which count covers with fixed arbitrary ramification over 0 and ∞ and simple ramification over b points, where b is given by the Riemann-Hurwitz formula. In this work, we introduce the notion of bi-pruned double

Following the breakthrough of Croot, Lev, and Pach [4], Tao [10] introduced a symmetrized version of their argument, which is now known as the slice rank method. In this paper, we introduce a more general version of the slice rank of a tensor, which we call the Partition Rank. This allows us to extend the slice rank method to problems that require the variables to be distinct. Using the partition rank

The Erdős-Ginzburg-Ziv constant of a finite abelian group G, denoted s(G), is the smallest k∈N such that any sequence of elements of G of length k contains a zero-sum subsequence of length exp⁡(G). In this paper, we use the partition rank from [14], which generalizes the slice rank, to prove that for any odd prime p,s(Fpn)≤(p−1)2p(J(p)⋅p)n where 0.84143, this is the first exponential improvement to

For each positive integer k, we consider five well-studied posets defined on the set of Dyck paths of semilength k. We prove that uniquely sorted permutations avoiding various patterns are equinumerous with intervals in these posets. While most of our proofs are bijective, some use generating trees and generating functions. We end with several conjectures.

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

In a recent paper, Ayyer and Behrend present for a wide class of partitions factorizations of Schur polynomials with an even number of variables where half of the variables are the reciprocals of the others into symplectic and/or orthogonal group characters, thereby generalizing results of Ciucu and Krattenthaler for rectangular shapes. Their proofs proceed by manipulations of determinants underlying

Let H be an induced subgraph of the torus Ckm. We show that when k≥3 is even and |V(H)| divides some power of k, then for sufficiently large n the torus Ckn has a perfect vertex-packing with induced copies of H. On the other hand, disproving a conjecture of Gruslys, we show that when k is odd and not a prime power, then there exists H such that |V(H)| divides some power of k, but there is no n such

The following long-standing problem in combinatorics was first posed in 1993 by Gessel and Reutenauer [5]. For which multisubsets B of the symmetric group Sn is the quasisymmetric functionQ(B)=∑π∈BFDes(π),n a symmetric function? Here Des(π) is the descent set of π and FDes(π),n is Gessel's fundamental basis for the vector space of quasisymmetric functions. The purpose of this paper is to provide a

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.

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 graph Γ is k-connected-homogeneous (k-CH) if k is a positive integer and any isomorphism between connected induced subgraphs of order at most k extends to an automorphism of Γ, and connected-homogeneous (CH) if this property holds for all k. Locally finite, locally connected graphs often fail to be 4-CH because of a combinatorial obstruction called the unique x property; we prove that this property

Vector parking functions are sequences of non-negative integers whose order statistics are bounded by a given integer sequence u=(u0,u1,u2,…). Using the theory of fractional power series and an analog of Newton-Puiseux Theorem, we derive the exponential generating function for the number of u-parking functions when u is periodic. Our method is to convert an Appell relation of Gončarov polynomials to

The linear representation of a subset of a finite projective space is an incidence system of affine points and lines determined by the subset. In this paper we use character theory to show that the rank of the incidence matrix has a direct geometric interpretation in terms of certain hyperplanes. We consider the LDPC codes defined by taking the incidence matrix and its transpose as parity-check matrices

Let P be a finite set of points in the plane in general position, that is, no three points of P are on a common line. We say that a set H of five points from P is a 5-hole in P if H is the vertex set of a convex 5-gon containing no other points of P. For a positive integer n, let h5(n) be the minimum number of 5-holes among all sets of n points in the plane in general position. Despite many efforts

The (classical) problem of characterizing and enumerating permutations that can be sorted using two stacks connected in series is still largely open. In the present paper we address a related problem, in which we impose restrictions both on the procedure and on the stacks. More precisely, we consider a greedy algorithm where we perform the rightmost legal operation. Moreover, the first stack is required

Buch, Kresch, Shimozono, Tamvakis, and Yong defined Hecke insertion to formulate a combinatorial rule for the expansion of the stable Grothendieck polynomials Gπ indexed by permutations in the basis of stable Grothendieck polynomials Gλ indexed by partitions. Patrias and Pylyavskyy introduced a shifted analogue of Hecke insertion whose natural domain is the set of maximal chains in a weak order on

We characterize non-singular polar spaces embedded in non-singular polar spaces of the same rank using subsets of generators satisfying a natural intersection condition. By [7] and [8] this result has applications to the theory of Cameron-Liebler sets. In fact, our result is a significant improvement of the main result in [7].

Two permutations of the natural numbers diverge if the absolute value of the difference of their elements in the same position goes to infinity. We show that there exists an infinite number of pairwise divergent permutations of the naturals. We relate this result to more general questions about the permutation capacity of infinite graphs.

We provide a characterization of the crystal bases for the quantum queer superalgebra recently introduced by Grantcharov et al. This characterization is a combination of local queer axioms generalizing Stembridge's local axioms for crystal bases for simply-laced root systems, which were recently introduced by Assaf and Oguz, with further axioms and a new graph G characterizing the relations of the

For fixed integers r>k≥2,e≥3, let fr(n,er−(e−1)k,e) be the maximum number of edges in an r-uniform hypergraph in which the union of any e distinct edges contains at least er−(e−1)k+1 vertices. A classical result of Brown, Erdős and Sós in 1973 showed that fr(n,er−(e−1)k,e)=Θ(nk). The degenerate Turán density is defined to be the limit (if it exists)π(r,k,e):=limn→∞⁡fr(n,er−(e−1)k,e)nk. Extending a

The Cheeger constant of a graph is the smallest possible ratio between the size of a subgraph and the size of its boundary. It is well known that this constant must be at least λ12, where λ1 is the smallest positive eigenvalue of the Laplacian matrix. The subject of this paper is a conjecture of the authors that for distance-regular graphs the Cheeger constant is at most λ1. In particular, we prove

It has been shown by Soprunov that the normalized mixed volume (minus one) of an n-tuple of n-dimensional lattice polytopes is a lower bound for the number of interior lattice points in the Minkowski sum of the polytopes. He defined n-tuples of mixed degree at most one to be exactly those for which this lower bound is attained with equality, and posed the problem of a classification of such tuples

We consider two analogues of a discrete version of the famous car parking problem of Rényi for trees, which are also known under the term random sequential adsorption. For both models, the blocking model, where cars arrive sequentially at the nodes and only park if the site and all neighbouring nodes are free, and the dimer model, where cars arrive sequentially at the edges and only park if both endnodes

We give a simple recursion labeled by binary sequences which computes rational q,t-Catalan power series, both in relatively prime and non relatively prime cases. It is inspired by, but not identical to recursions due to B. Elias, M. Hogancamp, and A. Mellit, obtained in their study of link homology. We also compare our recursion with the Hogancamp-Mellit's recursion and verify a connection between

Thus far, digraphs that are uniquely determined by their Hermitian spectra have proven elusive. Instead, researchers have turned to spectral determination of classes of switching equivalent digraphs, rather than individual digraphs. In the present paper, we consider the traditional notion: a digraph (or mixed graph) is said to be strongly determined by its Hermitian spectrum (abbreviated SHDS) if it

Binomial Eulerian polynomials first appeared in work of Postnikov, Reiner and Williams on the face enumeration of generalized permutohedra. They are γ-positive (in particular, palindromic and unimodal) polynomials which can be interpreted as h-polynomials of certain flag simplicial polytopes and which admit interesting Schur γ-positive symmetric function generalizations. This paper introduces analogues

We give a combinatorial formula for the Ehrhart h⁎-vector of the hypersimplex. In particular, we show that hd⁎(Δk,n) is the number of hypersimplicial decorated ordered set partitions of type (k,n) with winding number d, thereby proving a conjecture of N. Early. We do this by proving a more general conjecture of N. Early on the Ehrhart h⁎-vector of a generic cross-section of a hypercube.

Many combinatorial matrices — such as those of binomial coefficients, Stirling numbers of both kinds, and Lah numbers — are known to be totally non-negative, meaning that all minors (determinants of square submatrices) are non-negative. The examples noted above can be placed in a common framework: for each one there is a non-decreasing sequence (a1,a2,…), and a sequence (e1,e2,…), such that the (m

Let G(n,r,s) be a graph whose vertices are all r-element subsets of an n-element set, in which two vertices are adjacent if they intersect in exactly s elements. In this paper we study chromatic numbers of G(n,r,s) with r,s being fixed constants and n tending to infinity. Using a recent result of Keevash on existence of designs we deduce an inequality χ(G(n,r,s))⩽(1+o(1))nr−s(r−s−1)!(2r−2s−1)! for

For any r-graph H, we consider the problem of finding a rainbow H-factor in an r-graph G with large minimum ℓ-degree and an edge-colouring that is suitably bounded. We show that the asymptotic degree threshold is the same as that for finding an H-factor.

In this article we prove the e-positivity of Gν[X;q+1] when Gν[X;q] is a vertical strip LLT polynomial. This property has been conjectured in [2] and [7], and it implies several e-positivities conjectured in those references and in [3]. We make use of a result of Carlsson and Mellit [5] that shows that a vertical strip LLT polynomial can be obtained by applying certain compositions of operators of

We prove a “decomposition lemma” that allows us to count preimages of certain sets of permutations under West's stack-sorting map s. As a first application, we give a new proof of Zeilberger's formula for the number W2(n) of 2-stack-sortable permutations in Sn. Our proof generalizes, allowing us to find an algebraic equation satisfied by the generating function that counts 2-stack-sortable permutations

As a classical object, the Tamari lattice has many generalizations, including ν-Tamari lattices and parabolic Tamari lattices. In this article, we unify these generalizations in a bijective fashion. We first prove that parabolic Tamari lattices are isomorphic to ν-Tamari lattices for bounce paths ν. We then introduce a new combinatorial object called “left-aligned colorable tree”, and show that it

We introduce a graph structure on the set of Euclidean polytopes. The vertices of this graph are the d-dimensional polytopes contained in Rd and its edges connect any two polytopes that can be obtained from one another by either inserting or deleting a vertex, while keeping their vertex sets otherwise unaffected. We prove several results on the connectivity of this graph, and on a number of its subgraphs

We prove that the family of facets of a pure simplicial complex C of dimension up to three satisfies the Erdős-Ko-Rado property whenever C is flag and has no boundary ridges. We conjecture the same to be true in arbitrary dimension and give evidence for this conjecture. Our motivation is that complexes with these two properties include flag pseudo-manifolds and cluster complexes.

Using the concept of d-collapsibility from combinatorial topology, we define chordal simplicial complexes and show that their Stanley-Reisner ideals are componentwise linear. Our construction is inspired by and an extension of “chordal clutters” which was defined by Bigdeli, Yazdan Pour and Zaare-Nahandi in 2017, and characterizes Betti tables of all ideals with a linear resolution in a polynomial

Given a subset A⊆{0,1}n, let μ(A) be the maximal ratio between ℓ4 and ℓ2 norms of a function whose Fourier support is a subset of A.1 We make some simple observations about the connections between μ(A) and the additive properties of A on one hand, and between μ(A) and the uncertainty principle for A on the other hand. One application obtained by combining these observations with results in additive