In this paper, we establish a lower bound on the total number of inequivalent APN functions on the finite field with 22m elements, where m is even. We obtain this result by proving that the APN functions introduced by Pott and the second author [22], which depend on three parameters k, s and α, are pairwise inequivalent for distinct choices of the parameters k and s. Moreover, we determine the automorphism

A skew-morphism of a finite group G is a permutation σ on G fixing the identity element such that the product of 〈σ〉 with the left regular representation of G forms a permutation group on G. This permutation group is called the skew-product group of σ. The skew-morphism was introduced as an algebraic tool to investigate regular Cayley maps. In this paper, we characterize skew-products of skew-morphisms

The main goal of this paper is to give explicit descriptions of two maximal cones in the Gröbner fan of the Plücker ideal. These cones correspond to the monomial ideals given by semistandard and PBW-semistandard Young tableaux. For the first cone, as an intermediate result we obtain the description of a maximal cone in the Gröbner fan of any Hibi ideal. For the second, we generalize the notion of Hibi

We study lines through the origin of finite-dimensional complex vector spaces that enjoy a doubly transitive automorphism group. In doing so, we make fundamental connections with both discrete geometry and algebraic combinatorics. In particular, we show that doubly transitive lines are necessarily optimal packings in complex projective space, and we introduce a fruitful generalization of regular abelian

In this paper we show that the conjecture of Lemmens and Seidel of 1973 for systems of equiangular lines with common angle arccos⁡(1/5) is true. Our main tool is forbidden subgraphs for smallest Seidel eigenvalue −5.

We provide a homological algebraic realization of the lattices of integer cuts and integer flows of graphs. To a finite 2-edge-connected graph Γ with a spanning tree T, we associate a finite dimensional Koszul algebra AΓ,T. Under the construction, planar dual graphs with dual spanning trees are associated Koszul dual algebras. The Grothendieck group of the category of finitely-generated AΓ,T modules

There is a well-studied correspondence by Jaclyn Anderson between partitions that avoid hooks of length s or t and certain binary strings of length s+t. Using this map, we prove that the total size of a random partition of this kind converges in law to Watson's U2 distribution, as conjectured by Doron Zeilberger.

In the context of the (generalized) Delta Conjecture and its compositional form, D'Adderio, Iraci, and Vanden Wyngaerd recently stated a conjecture relating two symmetric function operators, Dk and Θk. We prove this Theta Operator Conjecture, finding it as a consequence of the five-term relation of Mellit and Garsia. As a result, we find surprising ways of writing the Dk operators. Even though we deal

Let G be connected uniform hypergraph and let A(G) be the adjacency tensor of G. The stabilizing index of G is exactly the number of eigenvectors of A(G) associated with the spectral radius, and the cyclic index of G is exactly the number of eigenvalues of A(G) with modulus equal to the spectral radius. Let G1⊙G2 and G1□G2 be the coalescence and Cartesian product of connected m-uniform hypergraphs

Two finite words u and v are called Abelian equivalent if each letter occurs equally many times in both u and v. The abelian closure A(x) of (the shift orbit closure of) an infinite word x is the set of infinite words y such that, for each factor u of y, there exists a factor v of x which is abelian equivalent to u. The notion of an abelian closure gives a characterization of Sturmian words: among

Murota (1998) and Murota and Shioura (1999) introduced concepts of M-convex function and M♮-convex function as discrete convex functions, which are generalizations of valuated matroids due to Dress and Wenzel (1992). In the present paper we consider a new operation defined by a convolution of sections of an M♮-convex function that transforms the given M♮-convex function to an M-convex function, which

In this paper we present a combinatorial proof of Selberg's integral formula. We prove a theorem about the number of topological orderings of a certain related directed graph bijectively. Selberg's integral formula then follows by induction. This solves a problem posed by R. Stanley in 2009. Our proof is based on Anderson's analytic proof of the formula. As part of the proof we show a further generalisation

Andrews and Newman have recently introduced the notion of the mex of a partition, the smallest positive integer that is not a part. The concept has been used since at least 2006, though, with connections to Frobenius symbols. Recently, the parity of the mex has been associated to the crank statistic named by Dyson in 1944. In this note, we extend and strengthen the connection between the crank and

Catalytic equations appear in several combinatorial applications, most notably in the enumeration of lattice paths and in the enumeration of planar maps. The main purpose of this paper is to show that the asymptotic estimate for the coefficients of the solutions of (so-called) positive catalytic equations has a universal asymptotic behavior. In particular, this provides a rationale why the number of

We introduce a reverse variant of the dual immaculate quasisymmetric functions, mirroring the dichotomy between quasisymmetric Schur functions and Young quasisymmetric Schur functions, and establish a lift of this basis to the polynomial ring. We show that taking stable limits of these reverse dual immaculate slide polynomials produces the reverse dual immaculate quasisymmetric functions, and we establish

Fully inhomogeneous spin Hall–Littlewood symmetric rational functions Fλ arise in the context of sl(2) higher spin six vertex models, and are multiparameter deformations of the classical Hall–Littlewood symmetric polynomials. We obtain a refined Cauchy identity expressing a weighted sum of the product of two Fλ's as a determinant. The determinant is of Izergin–Korepin type: it is the partition function

Let v be an odd real polynomial (i.e. a polynomial of the form ∑j=1ℓajx2j−1). We utilize sets of iterated differences to establish new results about sets of the form R(v,ϵ)={n∈N|‖v(n)‖<ϵ} where ‖⋅‖ denotes the distance to the closest integer. We then apply the new Diophantine results to obtain applications to ergodic theory and combinatorics. In particular, we obtain a new characterization of weakly

This note will give an enumeration of n-cycles in the symmetric group Sn by their degree (also known as their cyclic descent number) and studies similar counting problems for the conjugacy classes of n-cycles under the action of the rotation subgroup of Sn. This is achieved by relating such cycles to periodic orbits of an associated dynamical system acting on the circle. We also compute the mean and

In this article, a finite analogue of the generalized sum-of-tails identity of Andrews and Freitas is obtained. We derive several interesting results as special cases of this analogue, in particular, a recent identity of Dixit, Eyyunni, Maji and Sood. We derive a new extension of Abel's lemma with the help of which we obtain a one-parameter generalization of a sum-of-tails identity of Andrews, Garvan

Let n>k>t≥j≥1 be integers. Let X be an n-element set, (Xk) the collection of its k-subsets. A family F⊂(Xk) is called t-intersecting if |F∩F′|≥t for all F,F′∈F. The j'th shadow ∂jF is the collection of all (k−j)-subsets that are contained in some member of F. Estimating |∂jF| as a function of |F| is a widely used tool in extremal set theory. A classical result of the second author (Theorem 1.3) provides

A short, concise proof is given for that for k≥5 there exists a k-uniform hypergraph H without exponent, i.e., when the Turán function is not polynomial in n. More precisely, we have ex(n,H)=o(nk−1) but it exceeds nk−1−c for any positive c for n>n0(k,c). We conjecture that this is true for k∈{3,4} as well.

For a reduced word i of the longest element in the Weyl group of SLn+1(C), one can associate the string cone Ci which parametrizes the dual canonical bases. In this paper, we classify all i's such that Ci is simplicial. We also prove that for any regular dominant weight λ of sln+1(C), the corresponding string polytope Δi(λ) is unimodularly equivalent to the Gelfand–Cetlin polytope associated to λ if

Let G be a finite permutation group on Ω. An ordered sequence of elements of Ω, (ω1,…,ωt), is an irredundant base for G if the pointwise stabilizer G(ω1,…,ωt) is trivial and no point is fixed by the stabilizer of its predecessors. If all irredundant bases of G have the same size we say that G is an IBIS group. In this paper we show that if a primitive permutation group is IBIS, then it must be almost

Novák conjectured in 1974 that for any cyclic Steiner triple systems of order v with v≡1(mod6), it is always possible to choose one block from each block orbit so that the chosen blocks are pairwise disjoint. We consider the generalization of this conjecture to cyclic (v,k,λ)-designs with 1⩽λ⩽k−1. Superimposing multiple copies of a cyclic symmetric design shows that the generalization cannot hold for

We define, for each subset S of the set P of primes, an Sn-module LienS with interesting properties. Lien∅ is the well-known representation Lien of Sn afforded by the free Lie algebra, while LienP is the module Conjn of the conjugacy action of Sn on n-cycles. For arbitrary S the module LienS interpolates between the representations Lien and Conjn. We consider the symmetric and exterior powers of LienS

The quintuple product identity is deduced from the q-Dixon formula.

Davis and Jedwab (1997) established a great construction theory unifying many previously known constructions of difference sets, relative difference sets and divisible difference sets. They introduced the concept of building blocks, which played an important role in the theory. On the other hand, Polhill (2010) gave a construction of Paley type partial difference sets (conference graphs) based on a

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