In this paper, we prove a surprisingly simple formula that counts connected cycle-free families of set partitions, labelled free cacti and coloured Husimi graphs in which there are no blocks of the same colour that are incident to one another. We also provide a formula that enumerates noncrossing connected, cycle-free pairs of partitions.

We numerically investigate typical graphs in a region of the Strauss model of random graphs with constraints on the densities of edges and triangles. This region, where typical graphs had been expected to be bipodal but turned out to be tripodal, involves edge densities \(e\) below \(e_0 = (3-\sqrt{3})/6 \approx 0.2113\) and triangle densities \(t\) slightly below \(e^3\). We determine the extent of this region in \((e,t)\) space and show that there is a discontinuous phase transition at the boundary between this region and a bipodal phase. We further show that there is at least one phase transition within this region, where the parameters describing typical graphs change discontinuously.

This paper fits in the intersection between two disparate areas of combinatorics. Namely, graph theory and the combinatorics of Catalan words. A Catalan word with n parts is defined as a word w = w₁w₂⋯w_n over the set of positive integers in which w₁ = 1 and 1 ≤ w_k ≤ w_k − 1 + 1 for k = 2, 3, …, n. In order to study the intersection of the two areas, a specific type of graph called a grid graph is defined for each Catalan word. The main thrust of the paper is investigating the degrees of vertices in grid graphs. For each of the possible fixed degrees i ∈ {1, 2, 3, 4}, we find generating functions DF_i(x) where the coefficient of xⁿ is the total number of vertices of degree i in all grid graphs with n parts.

This note derives asymptotic upper and lower bounds for the number of planted plane trees on \(n\) nodes assigned labels from the set \(\{1, 2, \dots, k\}\) with the restriction that on any path from the root to a leaf, the labels must strictly decrease. We illustrate an application to calculating the largest eigenvalue of the adjacency matrix of a tree.

Let US be the class of all ultrametric spaces generated by labeled star graphs. We prove that compact US-spaces are the completions of totally bounded ultrametric spaces generated by decreasingly labeled rays. We characterize the ultrametric spaces which are weakly similar to finite US-spaces and describe these spaces by certain four-point conditions.

The Erdős-Anning Theorem states that an integer distance set in the Euclidean plane must have all of its points on a single line or is finite. However, this is not true if we consider area sets. That is, if \((x_1,y_1)\) and \((x_2,y_2)\) are any two vectors contained in the integer lattice, then the area of the parallelogram determined by the two vectors is an integer, showing that the points do not have to lie on a line. We prove a finite field version of these results for \(d=2\) and \(d=3\), showing that if \(E \subset \Bbb{F}_q^d, q=p^2\), where \(p\) is an odd prime and the distance set of \(E\) is \(\Bbb{F}_p\), then the size of \(E\) is at most \(p^d\). Furthermore, we prove that if the area set of \(E\) is a subset of \(\Bbb{F}_p\), then the size of \(E\) is at most \(p^2\) in two dimensions.

Let F_n denote the n-th Fibonacci number defined by F_n = F_n − 1 + F_n − 2 if n ≥ 2, with F₀ = 0 and F₁ = 1. In this paper, we find determinant identities for several Toeplitz–Hessenberg matrices whose nonzero entries are derived from the sequence k_n + m for various fixed m, where k_n = F_n − 1. These results may be obtained algebraically as special cases of more general formulas involving the Horadam numbers and the generating functions for the associated sequences of determinants. Equivalent multi-sum identities featuring sums of products of k_n terms with multinomial coefficients may be given, which follow from Trudi’s formula. Connections are made to several OEIS entries that have arisen previously in other contexts, perhaps most notably the Padovan number sequence. Finally, we provide combinatorial proofs of our identities involving k_n by enumerating (or finding the sum of signs of) various classes of tilings containing squares, dominos, trominos and a special type of tile which can be of arbitrary length.

Integer partitions of \( n \) are viewed as bargraphs (i.e., Ferrers diagrams rotated anticlockwise by 90 degrees) in which the \( i \)-th part of the partition \( x_i \) is given by the \( i \)-th column of the bargraph with \( x_i \) cells. The sun is at infinity in the northwest of our two-dimensional model, and each partition casts a shadow in accordance with the rules of physics. The number of unit squares in this shadow but not being part of the partition is found through a bivariate generating function in \( q \) tracking partition size and \( u \) tracking shadow. To do this, we define triangular \( q \)-binomial coefficients which are analogous to standard \( q \)-binomial coefficients, and we obtain a formula for these. This is used to obtain a generating function for the total number of shaded cells in (weakly decreasing)
partitions of \( n \).

We consider a scalar-valued implicit function of many variables, and provide two closed formulae for all of its partial derivatives. One formula is based on products of partial derivatives of the defining function, the other one involves fewer products of building blocks of multinomial type, and we study the combinatorics of the coefficients showing up in both formulae.

Tensors, or multi-linear forms, are important objects in a variety of areas from analytics, to combinatorics, to computational complexity theory. Notions of tensor rank aim to quantify the “complexity” of these forms, and are thus also important. While there is one single definition of rank that completely captures the complexity of matrices (and thus linear transformations), there is no definitive analog for tensors. Rather, many notions of tensor rank have been defined over the years, each with their own set of uses.

In this paper we survey the popular notions of tensor rank. We give a brief history of their introduction, motivating their existence, and discuss some of their applications in computer science. We also give proof sketches of recent results by Lovett, and Cohen and Moshkovitz, which prove asymptotic equivalence between three key notions of tensor rank over finite fields with at least three elements.