In 1995 Stanley introduced the chromatic symmetric function XG of a graph G, whose e-positivity and Schur-positivity has been of large interest. In this paper we study the relative e-positivity and Schur-positivity between connected graphs on n vertices. We define and investigate two families of posets on distinct chromatic symmetric functions. The relations depend on the e-positivity or Schur-positivity

A tanglegram is a pair of binary trees with the same set of leaves. Unlabeled tanglegrams were counted recently by Billey, Konvalinka, and Matsen, who also proposed the problem of counting several variations of unlabeled tanglegrams. We use the theory of combinatorial species to solve these problems.

A subfamily G⊆F⊆2[n] of sets is a non-induced (weak) copy of a poset P in F if there exists a bijection i:P→G such that p≤Pq implies i(p)⊆i(q). In the case where in addition p≤Pq holds if and only if i(p)⊆i(q), then G is an induced (strong) copy of P in F. We consider the minimum number sat(n,P) [resp. sat⁎(n,P)] of sets that a family F⊆2[n] can have without containing a non-induced [induced] copy

We study Young's seminormal basis vectors of the dual Specht modules of the symmetric group, indexed by a certain class of standard tableaux, and their denominators. These vectors include those whose denominators control the splitting of the canonical morphism Δ(λ+μ)→Δ(λ)⊗Δ(μ) over Z(p), where Δ(ν) is the Weyl module of the classical Schur algebra labelled by ν.

The Cayley graphs Gm(G)=Cay(Gm,S) of Cartesian powers of the finite group G with respect to canonical symmetric subsets S=S(m)⊂Gm are investigated. It is proved that for each m>1 the graph Gm(G) determines the group G up to isomorphism. The groups of automorphisms Aut(Gm(G)) are determined. It is shown that if G is a non-abelian group, then Aut(Gm(G))≃(Gm⋊Aut(G))⋊Dm+1, where Dm+1 is the dihedral group

The Weyl denominator identity has interesting combinatorial properties for several classes of Lie algebras. Along these lines, we prove that given a finite graph G, the chromatic symmetric function XG can be recovered from the Weyl denominator identity of a Borcherds-Kac-Moody Lie algebra g whose associated graph is G. This gives a connection between (a) the coefficients appearing when the chromatic

A set S⊂N of positive integers is a Sidon set if the pairwise sums of its elements are all distinct, or, equivalently, if|(x+w)−(y+z)|≥1 for every x,y,z,w∈S with x

In this paper we study variations of an old result by Müller, Reiterman, and the last author stating that a countable graph has a subgraph with infinite degrees if and only if in any labeling of the vertices (or edges) of this graph by positive integers one can always find an infinite increasing path. We study corresponding questions for simple hypergraphs and paths of infinite as well as of arbitrary

We construct a new partial geometry with parameters pg(5,5,2), not isomorphic to the partial geometry of van Lint and Schrijver.

Kühn, Osthus, and Treglown and, independently, Khan proved that if H is a 3-uniform hypergraph with n vertices, where n∈3Z and large, and δ1(H)>(n−12)−(2n/32), then H contains a perfect matching. In this paper, we show that for n∈3Z sufficiently large, if F1,…,Fn/3 are 3-uniform hypergraphs with a common vertex set and δ1(Fi)>(n−12)−(2n/32) for i∈[n/3], then {F1,…,Fn/3} admits a rainbow matching, i

We use the Chudnovsky-Seymour Real Root Theorem for independence polynomials to obtain some statements about the coefficients and roots of the Alexander and Conway polynomial of some types of plumbing links, addressing conjectures of Fox, Hoste and Liechti.

We prove that the enumerative polynomials of Stirling multipermutations by the statistics of plateaux, descents and ascents are partial γ-positive. Specialization of our result to the Jacobi-Stirling permutations confirms a recent partial γ-positivity conjecture due to Ma, Yeh and the second named author. Our partial γ-positivity expansion, as well as a combinatorial interpretation for the corresponding

We give a product formula for the number of shifted plane partitions of shifted double staircase shape with bounded entries. This is the first new example of a family of shapes with a plane partition product formula in many years. The proof is based on the theory of lozenge tilings; specifically, we apply the “free boundary” Kuo condensation due to Ciucu.

We extend recent results of Ono and Raji, relating the number of self-conjugate 7-core partitions to Hurwitz class numbers. Furthermore, we give a combinatorial explanation for the curious equality 2sc7(8n+1)=c4(7n+2). We also conjecture that an equality of this shape holds if and only if t=4, proving the cases t∈{2,3,5} and giving partial results for t>5.

We show that a set A of lines in PG(4,q), q even, is the set of secant lines of a parabolic (non-singular) quadric if and only if A satisfies the following three conditions: (I) every point of PG(4,q) lies on 0,12q3 or q3 lines of A; (II) every plane of PG(4,q) contains 0, 12q(q+1) or q2 lines of A; and (III) every hyperplane of PG(4,q) contains 12q2(q2+1), 12q3(q+1) or 12q2(q+1)2 lines of A.

In their work on the infinite flag variety, Lam, Lee, and Shimozono [30] introduced objects called bumpless pipe dreams and used them to give a formula for double Schubert polynomials. We extend this formula to the setting of K-theory, giving an expression for double Grothendieck polynomials as a sum over a larger class of bumpless pipe dreams. Our proof relies on techniques found in an unpublished

In this paper, we prove a congruence which contains a congruence conjectured by Adamchuk (OEIS A066796 in 2006, http://oeis.org/A066796). For any prime p≡1(mod3) and a∈Z+, we have∑k=123(pa−1)(2kk)≡0(modp2).

We provide a type-independent construction of an explicit basis for the semi-invariant harmonic differential forms of an arbitrary pseudo-reflection group in characteristic zero. Equivalently, we completely describe the structure of the χ-isotypic components of the corresponding super coinvariant algebras in one commuting and one anti-commuting set of variables, for all linear characters χ. In type

The moduli space M‾0,n may be embedded into the product of projective spaces P1×P2×⋯×Pn−3, using a combination of the Kapranov map |ψn|:M‾0,n→Pn−3 and the forgetful maps πi:M‾0,i→M‾0,i−1. We give an explicit combinatorial formula for the multidegree of this embedding in terms of certain parking functions of height n−3. We use this combinatorial interpretation to show that the total degree of the embedding

We consider the Norton algebra associated with a Q-polynomial primitive idempotent of the adjacency matrix for a distance-regular graph. We obtain a formula for the Norton algebra product that we find attractive.

In 2000, Kadell gave an orthogonality conjecture for a symmetric function generalization of the q-Dyson constant term identity or the Zeilberger–Bressoud q-Dyson theorem. The non-zero part of Kadell's orthogonality conjecture is a constant term identity indexed by a weak composition v=(v1,…,vn) in the case when only one vi≠0. This conjecture was first proved by Károlyi, Lascoux and Warnaar in 2015

We study and classify faithfully balanced modules for the algebra of triangular n by n matrices and more generally for Nakayama algebras. The theory extends known results about tilting modules, which are classified by binary trees, and counted with the Catalan numbers. The number of faithfully balanced modules is a 2-factorial number. Among them are n! modules with n indecomposable summands, which

We propose periodic Macdonald processes as a (q,t)-deformation of periodic Schur processes and a periodic analogue of Macdonald processes. It is known that, in the theory of stochastic processes related to a family of symmetric functions, the Cauchy-like identity gives an explicit expression of a partition function. We compute the partition functions of periodic Macdonald processes relying on the free

A permutation σ∈Sn is said to be k-universal or a k-superpattern if for every π∈Sk, there is a subsequence of σ that is order-isomorphic to π. A simple counting argument shows that σ can be a k-superpattern only if n≥(1/e2+o(1))k2, and Arratia conjectured that this lower bound is best-possible. Disproving Arratia's conjecture, we improve the trivial bound by a small constant factor. We accomplish this

A set of integers S⊂N is an α–strong Sidon set if the pairwise sums of its elements are far apart by a certain measure depending on α, more specifically if|(x+w)−(y+z)|≥max⁡{xα,yα,zα,wα} for every x,y,z,w∈S satisfying max⁡{x,w}≠max⁡{y,z}. We obtain a new lower bound for the growth of α–strong infinite Sidon sets when 0≤α<1. We also further extend that notion in a natural way by obtaining the first

We prove several relations on the f-vectors and Betti numbers of flag complexes. For every flag complex Δ, we show that there exists a balanced complex with the same f-vector as Δ, and whose top-dimensional Betti number is at least that of Δ, thereby extending a theorem of Frohmader by additionally taking homology into consideration. We obtain upper bounds on the top-dimensional Betti number of Δ in

Motivated by a question of R. Nandakumar, we show that the Euclidean plane can be dissected into mutually incongruent convex quadrangles of the same area and the same perimeter. As a byproduct we obtain vertex-to-vertex dissections of the plane by mutually incongruent triangles of unit area that are arbitrarily close to the periodic vertex-to-vertex tiling by equilateral triangles.

If 2≤d≤k and n≥dk/(d−1), a d-cluster is defined to be a collection of d elements of ([n]k) with empty intersection and union of size no more than 2k. Mubayi [14] conjectured that the largest size of a d-cluster-free family F⊂([n]k) is (n−1k−1), with equality holding only for a maximum-sized star. Here, we resolve Mubayi's conjecture and prove a slightly stronger result, thus completing a new generalization

With the help of the creative microscoping method recently introduced by Guo and Zudilin and the Chinese remainder theorem for coprime polynomials, we establish a q-supercongruence with two parameters modulo [n]Φn(q)3. Here [n]=(1−qn)/(1−q) and Φn(q) is the n-th cyclotomic polynomial in q. In particular, we confirm a recent conjecture of Guo and give a complete q-analogue of Long's supercongruence

Given a family F⊂2[n], its diversity is the number of sets not containing an element with the highest degree. The concept of diversity has proven to be very useful in the context of k-uniform intersecting families. In this paper, we study (different notions of) diversity in the context of other extremal set theory problems. One of the main results of the paper is a sharp stability result for cross-intersecting

We give an explicit, nonnegative formula for the expansion of nonsymmetric Macdonald polynomials specialized at t=0 in terms of Demazure characters. Our formula results from constructing Demazure crystals whose characters are the nonsymmetric Macdonald polynomials, which also gives a new proof that these specialized nonsymmetric Macdonald polynomials are positive graded sums of Demazure characters

Motivated by coding applications, two enumeration problems are considered: the number of distinct divisors of a degree-m polynomial over F=GF(q), and the number of ways a polynomial can be written as a product of two polynomials of degree at most n over F. For the two problems, bounds are obtained on the maximum number of factorizations, and a characterization is presented for polynomials attaining

The binomial Eulerian polynomials, first introduced in work of Postnikov, Reiner and Williams, are γ-positive polynomials and can be interpreted as h-polynomials of certain flag simplicial polytopes. Recently, Athanasiadis studied analogs of these polynomials for colored permutations and proved that they can be written as the sums of two γ-positive polynomials. In this paper, we find combinatorial

We prove several exact quantitative versions of Helly's and Tverberg's theorems, which guarantee that a finite family of convex sets in Rd has a large intersection. Our results characterize conditions that are sufficient for the intersection of a family of convex sets to contain a “witness set” which is large under some concave or log-concave measure. The possible witness sets include ellipsoids, zonotopes

Let G be the group scheme SLd+1 over Z and let Q be the parabolic subgroup scheme corresponding to the simple roots α2,⋯,αd−1. Then G/Q is the Z-scheme of partial flags {D1⊂Hd⊂V}. We will calculate the cohomology modules of line bundles over this flag scheme. We will prove that the only non-trivial ones are isomorphic to the kernel or the cokernel of certain matrices with multinomial coefficients.

Given two non-negative integers n and s, define m(n,s) to be the maximal number such that in every hypergraph H on n vertices and with at most m(n,s) edges there is a vertex x such that |Hx|≥|E(H)|−s, where Hx={H∖{x}:H∈E(H)}. This problem has been posed by Füredi and Pach and by Frankl and Tokushige. While the first results were only for specific small values of s, Frankl determined m(n,2d−1−1) for

This paper studies combinatorial properties of the complex of planar injective words, a chain complex of modules over the Temperley-Lieb algebra that arose in our work on homological stability. Despite being a linear rather than a discrete object, our chain complex nevertheless exhibits interesting combinatorial properties. We show that the Euler characteristic of this complex is the n-th Fine number

In 2008 Brändén proved a (p,q)-analogue of the γ-expansion formula for Eulerian polynomials and conjectured the divisibility of the γ-coefficient γn,k(p,q) by (p+q)k. As a follow-up, in 2012 Shin and Zeng showed that the fraction γn,k(p,q)/(p+q)k is a polynomial in N[p,q]. The aim of this paper is to give a combinatorial interpretation of the latter polynomial in terms of André permutations, a class

Recently, it was proved by Anari–Oveis Gharan–Vinzant, Anari–Liu–Oveis Gharan–Vinzant and Brändén–Huh that, for any matroid M, its basis generating polynomial and its independent set generating polynomial are log-concave on the positive orthant. Using these, they obtain some combinatorial inequalities on matroids including a solution of strong Mason's conjecture. In this paper, we study the strictness

A subspace of F2n is called cyclically covering if every vector in F2n has a cyclic shift which is inside the subspace. Let h2(n) denote the largest possible codimension of a cyclically covering subspace of F2n. We show that h2(p)=2 for every prime p such that 2 is a primitive root modulo p, which, assuming Artin's conjecture, answers a question of Peter Cameron from 1991. We also prove various bounds

Let K be an abelian group of order v. A Steiner quadruple system of order v (SQS(v)) (K,B) is called symmetric K-invariant if for each B∈B, it holds that B+x∈B for each x∈K and B=−B+y for some y∈K. When the Sylow 2-subgroup of K is cyclic, a necessary and sufficient condition for the existence of a symmetric K-invariant SQS(v) was given by Munemasa and Sawa, which is a generalization of a necessary

We generalize the Tutte polynomial of a matroid to a morphism of matroids via the K-theory of flag varieties. We introduce two different generalizations, and demonstrate that each has its own merits, where the trade-off is between the ease of combinatorics and geometry. One generalization recovers the Las Vergnas Tutte polynomial of a morphism of matroids, which admits a corank-nullity formula and

Let a and h be positive integers and let p be a prime. Let q1,…,qt be the distinct prime divisors of h and write Q(h)={∑i=1tciqi:ci∈Z,ci≥0}. We provide constructions of group invariant Butson Hadamard matrices BH(G,h) in the following cases. 1. G=(Zp)2a and at least one of the following conditions is satisfied. • pa∈Q(h), • pa+2∈Q(h) and h is even, • pa+1=(q1−1)(q2−1) where q1 and q2 are distinct prime

We give a new formula for the branching rule from GLn to On generalizing the Littlewood's restriction formula. The formula is given in terms of Littlewood-Richardson tableaux with certain flag conditions which vanish in a stable range. As an application, we give a combinatorial formula for the Lusztig t-weight multiplicity Kμ0(t) of type Bn and Dn with highest weight μ and weight 0.

We prove canonical and non-canonical tree-of-tangles theorems for abstract separation systems that are merely structurally submodular. Our results imply all known tree-of-tangles theorems for graphs, matroids and abstract separation systems with submodular order functions, with greatly simplified and shortened proofs.

The geometric dual of a cellularly embedded graph is a fundamental concept in graph theory and also appears in many other branches of mathematics. The partial dual is an essential generalization which can be obtained by forming the geometric dual with respect to only a subset of edges of a cellularly embedded graph. Twisted duality is a further generalization from combining partial Petrials with partial

Stirling permutations were introduced by Gessel and Stanley in 1978, who enumerated them by the number of descents to give a combinatorial interpretation of certain polynomials related to Stirling numbers. A natural extension of these permutations are quasi-Stirling permutations, which can be viewed as labeled noncrossing matchings. They were recently introduced by Archer et al., motivated by the fact

Ramanujan's modular equations of prime degrees 3,5,7,11 and 23 are associated with elegant colored partition theorems. In 2005, S. O. Warnaar established a general identity which implies the modular equations of degrees 3 and 7. In this paper, we provide a generalization of the remaining modular equations of degrees 5,11 and 23. In addition, we derive many partition identities from the generalization

We obtain a finite analogue of a recent generalization of an identity in Ramanujan's Notebooks. Differentiating it with respect to one of the parameters leads to a result whose limiting case gives a finite analogue of Andrews' famous identity for spt(n). The latter motivates us to extend the theory of the restricted partition function p(n,N), namely, the number of partitions of n with largest parts

We investigate sc7(n), the number of self-conjugate 7-core partitions of size n. It turns out that sc7(n)=0 for n≡7(mod8). For n≡1,3,5(mod8), with n≢5(mod7), we find that sc7(n) is essentially a Hurwitz class number. Using recent work of Gao and Qin, we show thatsc7(n)=2−ε(n)−1⋅H(−Dn), where −Dn:=−4ε(n)(7n+14) and ε(n):=12⋅(1+(−1)n−12). This fact implies several corollaries which are of interest. For

Symplectic Q-functions are a symplectic analogue of Schur Q-functions and defined as the t=−1 specialization of Hall–Littlewood functions associated with the root system of type C. In this paper we prove that symplectic Q-functions share many of the properties of Schur Q-functions, such as a tableau description and a Pieri-type rule. And we present some positivity conjectures, including the positivity

A direct saddle-point analysis (without relying on any modular forms or functional equations) is developed to establish the asymptotics of Fishburn matrices and a large number of other variants with a similar sum-of-finite-product form for their (formal) generating functions. In addition to solving some conjectures, the application of our saddle-point approach to the distributional aspects of statistics

In this erratum we explain how to implement two axioms stated in our paper of 2009 so as to get a purely “local” characterization for finite B2-crystals, which was declared but not clarified at some moments there. Also we correct some inaccuracies in that paper.

We classify recurrent states of the Abelian sandpile model (ASM) on the complete split graph. There are two distinct cases to be considered that depend upon the location of the sink vertex in the complete split graph. This characterisation of decreasing recurrent states is in terms of Motzkin words and can also be characterised in terms of combinatorial necklaces. We also give a characterisation of

In 2003, Alladi, Andrews and Berkovich proved an identity for partitions where parts occur in eleven colors: four primary colors, six secondary colors, and one quaternary color. Their work answered a longstanding question of how to go beyond a classical theorem of Göllnitz, which uses three primary and three secondary colors. Their main tool was a deep and difficult four parameter q-series identity

We give simple arithmetic conditions that force the Sylow p-subgroup of the critical group of a strongly regular graph to take a specific form. These conditions depend only on the parameters (v,k,λ,μ) of the strongly regular graph under consideration. We give many examples, including how the theory can be used to compute the critical group of Conway's 99-graph and to give an elementary argument that

A recent conjecture that appeared in three papers by Bigdeli–Faridi, Dochtermann, and Nikseresht, is that every simplicial complex whose clique complex has shellable Alexander dual, is ridge-chordal. This strengthens the long-standing Simon's conjecture that the k-skeleton of the simplex is extendably shellable, for any k. We show that the stronger conjecture has a negative answer, by exhibiting an

In a recent paper, Bruns and von Thaden established a bound for the length of vectors involved in a unimodular triangulation of simplicial cones. The bound is exponential in the square of the logarithm of the multiplicity, and improves previous bounds significantly. In this paper we will prove that a bound, which is polynomial in the multiplicity μ, exists. In detail, the bound is of the type μf(d)

In 2007 Lam and Pylyavskyy found a combinatorial formula for the dual stable Grothendieck polynomials, which are the dual basis of the stable Grothendieck polynomials with respect to the Hall inner product. In 2016 Galashin, Grinberg, and Liu introduced refined dual stable Grothendieck polynomials by putting additional sequence of parameters in the combinatorial formula of Lam and Pylyavskyy. Grinberg

The index conjecture in zero-sum theory states that if gcd⁡(n,6)=1 then every minimal zero-sum sequence of length 4 modulo n has index 1. In this paper, we prove a version of this conjecture for all n>N for some absolute constant N.