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.