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 ring. We show d-collapsible and d-representable complexes produce componentwise linear ideals for appropriate d. Along the way, we prove that there are generators that when added to the ideal, do not change Betti numbers in certain degrees. We then show that large classes of componentwise linear ideals, such as Gotzmann ideals and square-free stable ideals have chordal Stanley-Reisner complexes, that Alexander duals of vertex decomposable complexes are chordal, and conclude that the Betti table of every componentwise linear ideal is identical to that of the Stanley-Reisner ideal of a chordal complex.

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 number theory is a stability result for the uncertainty principle on the discrete cube. Our more technical contribution is determining μ(A) rather precisely, when A is a Hamming sphere S(n,k) for all 0≤k≤n.

Many natural notions of additive and multiplicative largeness arise from results in Ramsey theory. In this paper, we explain the relationships between these notions for subsets of N and in more general ring-theoretic structures. We show that multiplicative largeness begets additive largeness in three ways and give a collection of examples demonstrating the optimality of these results. We also give a variety of applications arising from the connection between additive and multiplicative largeness. For example, we show that given any n,k∈N, any finite set with fewer than n elements in a sufficiently large finite field can be translated so that each of its elements becomes a non-zero kth power. We also prove a theorem concerning Diophantine approximation along multiplicatively syndetic subsets of N and a theorem showing that subsets of positive upper Banach density in certain multiplicative sub-semigroups of N of zero density contain arbitrarily long arithmetic progressions. Along the way, we develop a new characterization of upper Banach density in a wide class of amenable semigroups and make explicit the uniformity in recurrence theorems from measure theoretic and topological dynamics. This in turn leads to strengthened forms of classical theorems of Szemerédi and van der Waerden on arithmetic progressions.

We express minors of Toeplitz matrices of finite and large dimension in terms of symmetric functions. Comparing the resulting expressions with the inverses of some Toeplitz matrices, we obtain explicit formulas for a Selberg-Morris integral and for specializations of certain skew Schur polynomials.

In this paper we explore the asymptotic enumeration of three-dimensional excursions confined to the positive octant. As shown in [29], both the exponential growth and the critical exponent admit universal formulas, respectively in terms of the inventory of the step set and of the principal Dirichlet eigenvalue of a certain spherical triangle, itself being characterized by the steps of the model. We focus on the critical exponent, and our main objective is to relate combinatorial properties of the step set (structure of the so-called group of the walk, existence of a Hadamard decomposition, existence of differential equations satisfied by the generating functions) to geometric or analytic properties of the associated spherical triangle (remarkable angles, tiling properties, existence of an exceptional closed-form formula for the principal eigenvalue). As in general the eigenvalues of the Dirichlet problem on a spherical triangle are not known in closed form, we also develop a finite-elements method to compute approximate values, typically with ten digits of precision.

Partial difference sets (for short, PDSs) with parameters (n2, r(n−ϵ), ϵn+r2−3ϵr, r2−ϵr) are called Latin square type (respectively negative Latin square type) PDSs if ϵ=1 (respectively ϵ=−1). In this paper, we will give restrictions on the parameter r of a (negative) Latin square type partial difference set in an abelian group of non-prime power order a2b2, where gcd⁡(a,b)=1, a>1, and b is an odd positive integer ≥3. Very few general restrictions on r were previously known. Our restrictions are particularly useful when a is much larger than b. As an application, we show that if there exists an abelian negative Latin square type PDS with parameter set (9p4s,r(3p2s+1),−3p2s+r2+3r,r2+r), 1≤r≤3p2s−12, p≡1(mod4) a prime number and s is an odd positive integer, then there are at most three possible values for r. For two of these three r values, J. Polhill gave constructions in 2009 [10].

Motivated by the classical Eulerian triangle and triangular arrays from staircase tableaux and tree-like tableaux, we study a generalized Eulerian array [Tn,k]n,k≥0, which satisfies the recurrence relation:Tn,k=λ(a1k+a2)Tn−1,k+[(b1−da1)n−(b1−2da1)k+b2−d(a1−a2)]Tn−1,k−1+d(b1−da1)λ(n−k+1)Tn−1,k−2, where T0,0=1 and Tn,k=0 unless 0≤k≤n. We derive some properties of [Tn,k]n,k≥0, including the explicit formulae of Tn,k and the exponential generating function of the generalized Eulerian polynomial Tn(q), and the ordinary generating function of Tn(q) in terms of the Jacobi continued fraction expansion, and real rootedness and log-concavity of Tn(q), stability of the iterated Turán-type polynomial Tn+1(q)Tn−1(q)−Tn2(q). Furthermore, we also prove the q-Stieltjes moment property and 3-q-log-convexity of Tn(q) and that the triangular convolution preserves Stieltjes moment property of sequences. In addition, we also give a criterion for γ-positivity in terms of the Jacobi continued fraction expansion. In consequence, we get γ-positivity of a generalized Narayana polynomial, which implies that of Narayana polynomials of types A and B in a unified manner. We also derive γ-positivity for a symmetric sub-array of [Tn,k]n,k≥0, which in particular gives a unified proof of γ-positivity of Eulerian polynomials of types A and B. Our results not only can immediately apply to Eulerian triangles of two kinds and arrays from staircase tableaux and tree-like tableaux, but also to segmented permutations and flag excedance numbers in colored permutations groups in a unified approach. In particular, we also confirm a conjecture of Nunge about the unimodality from segmented permutations.

Let □n denote the folded n-cube and let A(□n,d) denote the maximum size of a code in □n with minimum distance at least d. We give an upper bound on A(□n,d) based on block-diagonalizing the Terwilliger algebra of □n and on semidefinite programming. The technique of this paper is an extension of the approach taken by A. Schrijver [11] on the study of upper bounds for binary codes.

In this paper, we prove Erdős-Ko-Rado type results for (a) family of set partitions where the size of each block is a multiple of k; and (b) family of set partitions with minimum block size k.

Quantum walks, an important tool in quantum computing, have been very successfully investigated using techniques in algebraic graph theory. We are motivated by the study of state transfer in continuous-time quantum walks, which is understood to be a rare and interesting phenomenon. We consider a perturbation on an edge uv of a graph where we add a weight β to the edge and a loop of weight γ to each of u and v. We characterize when this perturbation results in strongly cospectral vertices u and v. Applying this to strongly regular graphs, we give infinite families of strongly regular graphs where some perturbation results in perfect state transfer. Further, we show that, for every strongly regular graph, there is some perturbation which results in pretty good state transfer. We also show for any strongly regular graph X and edge e∈E(X), that ϕ(X\e) does not depend on the choice of e.

The enumeration of planar maps equipped with an Eulerian orientation has attracted attention in both combinatorics and theoretical physics since at least 2000. The case of 4-valent maps is particularly interesting: these orientations are in bijection with properly 3-coloured quadrangulations, while in physics they correspond to configurations of the ice model. We solve both problems – namely the enumeration of planar Eulerian orientations and of 4-valent planar Eulerian orientations – by expressing the associated generating functions as the inverses (for the composition of series) of simple hypergeometric series. Using these expressions, we derive the asymptotic behaviour of the number of planar Eulerian orientations, thus proving earlier predictions of Kostov, Zinn-Justin, Elvey Price and Guttmann. This behaviour, μn/(nlog⁡n)2, prevents the associated generating functions from being D-finite. Still, these generating functions are differentially algebraic, as they satisfy non-linear differential equations of order 2. Differential algebraicity has recently been proved for other map problems, in particular for maps equipped with a Potts model. Our solutions mix recursive and bijective ingredients. In particular, a preliminary bijection transforms our oriented maps into maps carrying a height function on their vertices. In the 4-valent case, we also observe an unexpected connection with the enumeration of maps equipped with a spanning tree that is internally inactive in the sense of Tutte. This connection remains to be explained combinatorially.

The supercharacter theory of algebra groups gave us a representation theoretic realization of the Hopf algebra of symmetric functions in noncommuting variables. The underlying representation theoretic framework comes equipped with two canonical bases, one of which was completely new in terms of symmetric functions. This paper simultaneously generalizes this Hopf structure by considering a larger class of groups while also restricting the representation theory to a more combinatorially tractable one. Using the normal lattice supercharacter theory of pattern groups, we not only gain a third canonical basis, but also are able to compute numerous structure constants in the corresponding Hopf monoid, including coproducts and antipodes for the new bases.

A compacted binary tree is a graph created from a binary tree such that repeatedly occurring subtrees in the original tree are represented by pointers to existing ones, and hence every subtree is unique. Such representations form a special class of directed acyclic graphs. We are interested in the asymptotic number of compacted trees of given size, where the size of a compacted tree is given by the number of its internal nodes. Due to its superexponential growth this problem poses many difficulties. Therefore we restrict our investigations to compacted trees of bounded right height, which is the maximal number of edges going to the right on any path from the root to a leaf. We solve the asymptotic counting problem for this class as well as a closely related, further simplified class. For this purpose, we develop a calculus on exponential generating functions for compacted trees of bounded right height and for relaxed trees of bounded right height, which differ from compacted trees by dropping the above described uniqueness condition. This enables us to derive a recursively defined sequence of differential equations for the exponential generating functions. The coefficients can then be determined by performing a singularity analysis of the solutions of these differential equations. Our main results are the computation of the asymptotic numbers of relaxed as well as compacted trees of bounded right height and given size, when the size tends to infinity.

A method is described to count simple diagonal walks on Z2 with a fixed starting point and endpoint on one of the axes and a fixed winding angle around the origin. The method involves the decomposition of such walks into smaller pieces, the generating functions of which are encoded in a commuting set of Hilbert space operators. The general enumeration problem is then solved by obtaining an explicit eigenvalue decomposition of these operators involving elliptic functions. By further restricting the intermediate winding angles of the walks to some open interval, the method can be used to count various classes of walks restricted to cones in Z2 of opening angles that are integer multiples of π/4. We present three applications of this main result. First we find an explicit generating function for the walks in such cones that start and end at the origin. In the particular case of a cone of angle 3π/4 these walks are directly related to Gessel's walks in the quadrant, and we provide a new proof of their enumeration. Next we study the distribution of the winding angle of a simple random walk on Z2 around a point in the close vicinity of its starting point, for which we identify discrete analogues of the known hyperbolic secant laws and a probabilistic interpretation of the Jacobi elliptic functions. Finally we relate the spectrum of one of the Hilbert space operators to the enumeration of closed loops in Z2 with fixed winding number around the origin.

Harper's Theorem states that, in a hypercube, among all sets of a given fixed size the Hamming balls have minimal closed neighbourhoods. In this paper we prove a stability-like result for Harper's Theorem: if the closed neighbourhood of a set is close to minimal in the hypercube, then the set must be very close to a Hamming ball around some vertex.