We consider the number of common edges in two independent random spanning trees of a graph G. For complete graphs Kn, we give a new proof of the fact, originally obtained by Moon, that the distribution converges to a Poisson distribution with expected value 2. This is applied to show a Poisson limit law for the number of common edges in two independent random spanning trees of an Erdős–Rényi random graph G(n,p) for constant p, as well as a central limit theorem in the case where p→0 and p≥n−2/3+ɛ. We also use the same method to prove an analogous result for complete multipartite graphs.

In this paper, we study the asymptotic behaviour of the number of subtrees and the subtree density for a sequence of trees that converges in the Benjamini-Schramm sense. Benjamini-Schramm convergence, also called local weak convergence, describes the local behaviour of a sequence of graphs. Here we show that for a Benjamini-Schramm-convergent sequence of trees, the subtree entropy, i.e., the logarithm of the number of subtrees divided by the order, converges to a constant depending only on the limit. The same holds true for the subtree density, i.e., the probability of a uniformly random vertex being contained in a uniformly random subtree, provided that long paths are ruled out in the limit. Related to this, we show that the subtree density and the average subtree entropy are dense in different parts of the unit interval [0,1] for both general trees and series-reduced trees.

A B-tree is a type of search tree where every node (except possibly for the root) contains between m and 2m keys for some positive integer m, and all leaves have the same distance to the root. We study sequences of B-trees that can arise from successively inserting keys, and in particular present a bijection between such sequences (which we call histories) and a special type of increasing trees. We describe the set of permutations for the keys that belong to a given history, and also show how to use this bijection to analyse statistics associated with B-trees.

A binary tree (more precisely, an unrooted binary tree) is a tree in which all internal vertices (i.e., non-leaves) are exactly of degree 3. We give an upper bound and a lower bound for the number of maximal independent sets in binary trees together with a characterization of the extremal binary trees. The binary trees with second largest number of maximal independent sets are also characterized. (c) 2025 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

In this paper we present new results on the enumeration of parking functions and labeled forests. We introduce new statistics on parking functions, which are then extended to labeled forests via bijective correspondences. We determine the joint distribution of two statistics on parking functions and their counterparts on labeled forests. Our results on labeled forests also serve to explain the mysterious equidistribution between two seemingly unrelated statistics in parking functions recently identified by Stanley and Yin and give an explicit bijection between the two statistics. Extensions of our techniques are discussed, including joint distribution on further refinement of these new statistics.

We consider the enumeration of plane trees (rooted ordered trees) whose vertices are colored according to a specific coloring rule that prescribes which possible pairs of colors can occur as the colors of a parent vertex and its child. This general construction covers many different examples that have been studied in the literature. Some general necessary and sufficient conditions for two different coloring rules to result in the same counting sequence are established. We also provide exhaustive lists of counting sequences arising from coloring rules with two or three colors, and we find formulas and closed form expressions for many of these sequences. The famous Fibonacci, Catalan, Narayana, and Schröder sequences appear in several cases. Some of these coloring rules are extended to families of coloring rules with arbitrarily many colors.

Making use of a newly developed package in the computer mathematics system SageMath, we show how to perform a full asymptotic analysis of certain types of sums that occur frequently in combinatorics, including explicit error bounds. We present two applications of the general approach to illustrate its use: the first concerns a classical problem due to Ramanujan, while the second one concerns a question of B & oacute;na and DeJonge on 132-avoiding permutations with a unique longest increasing subsequence that can be translated into an inequality for a certain binomial sum.

For a graph G with n vertices and a positive integer k ≤ n, let s_k(G) be the number of subtrees (subgraphs that are trees, not necessarily induced) of G with k vertices. The subtree polynomial of G is S(G;x) = ∑_k=1^n s_k(G) x^k. In this paper, we consider dense connected graphs with a minimum degree that is linear in the number of vertices. We prove that the number of missing vertices in a random subtree is asymptotically Poisson-distributed and deduce that all the roots of the subtree polynomial have to be close to 0.

In this paper we study cycles in multiset permutations and parking functions. As combinatorial objects, multiset permutations are essential building blocks for mappings and permutations, while parking functions lie between mappings and permutations. We take both algebraic and analytic views in our investigation and present exact as well as asymptotic results. We point to a surprising correspondence between two statistics on multiset permutations, terminal closers and cyclic points, shedding light on the combinatorial structure.

We characterize the extremal trees that maximize the number of almost-perfect matchings, which are matchings covering all but one or two vertices, and those that maximize the number of strong almost-perfect matchings, which are matchings missing only one or two leaves. We also determine the trees that minimize the number of maximal matchings. We apply these results to extremal problems on the weighted Hosoya index for several choices of vertex-degree-based weight function.

An (unrooted) d-ary tree is a tree in which every internal vertex has degree d+1 . In this paper, we show for every fixed d≥ 2 that d-ary caterpillars have the minimum number of dominating sets among d-ary trees of a given order. We also determine the maximum number of dominating sets in binary trees (the special case d=2 ) and classify the extremal trees, which are also unique.

It has been conjectured that the minimum and maximum of the mean subtree order among connected graphs of order n are attained by the path P_n and clique K_n, respectively. Extending ideas due to Haslegrave and Vince, we confirm that the minimum is indeed attained by P_n. On the other hand, we discuss different approaches (both promising and flawed) that could lead to a proof of the extremality of K_n.

Motivated by a question and some enumerative conjectures of Richard Stanley, we explore the equivalence classes of words in the Weyl algebra, 𝐤⟨ D,U⟩/(DU-UD=1). We show that each class is generated by the swapping of adjacent *balanced subwords*, i.e., those which have the same number of D's as U's, and give several other characterizations. Armed with this we deduce a number of enumerative results about the number of such equivalence classes and their sizes. We extend these results to the class of c-Dyck words, where every prefix has at least c times as many U's as D's. We also connect these results to previous work on bond percolation and rook theory, and generalize them to some other algebras.

Random directed graphs $D(n,p)$ undergo a phase transition around the point $p = 1/n$, and the width of the transition window has been known since the works of Luczak and Seierstad. They have established that as $n \to \infty$ when $p = (1 + \mu n^{-1/3})/n$, the asymptotic probability that the strongly connected components of a random directed graph are only cycles and single vertices decreases from 1 to 0 as $\mu$ goes from $-\infty$ to $\infty$. By using techniques from analytic combinatorics, we establish the exact limiting value of this probability as a function of $\mu$ and provide more properties of the structure of a random digraph around, below and above its transition point. We obtain the limiting probability that a random digraph is acyclic and the probability that it has one strongly connected complex component with a given difference between the number of edges and vertices (called excess). Our result can be extended to the case of several complex components with given excesses as well in the whole range of sparse digraphs. Our study is based on a general symbolic method which can deal with a great variety of possible digraph families, and a version of the saddle point method which can be systematically applied to the complex contour integrals appearing from the symbolic method. While the technically easiest model is the model of random multidigraphs, in which multiple edges are allowed, and where edge multiplicities are sampled independently according to a Poisson distribution with a fixed parameter $p$, we also show how to systematically approach the family of simple digraphs, where multiple edges are forbidden, and where 2-cycles are either allowed or not. Our theoretical predictions are supported by numerical simulations, and we provide tables of numerical values for the integrals of Airy functions that appear in this study.

The problem of determining the maximum number of maximal independent sets in certain graph classes dates back to a paper of Miller and Muller and a question of Erd\H{o}s and Moser from the 1960s. The minimum was always considered to be less interesting due to simple examples such as stars. In this paper we show that the problem becomes interesting when restricted to twin-free graphs, where no two vertices have the same open neighbourhood. We consider the question for arbitrary graphs, bipartite graphs and trees. The minimum number of maximal independent sets turns out to be logarithmic in the number of vertices for arbitrary graphs, linear for bipartite graphs and exponential for trees. In the latter case, the minimum and the extremal graphs have been determined earlier by Taletski\u{\i} and Malyshev, but we present a shorter proof.

A fringe subtree of a rooted tree is a subtree that consists of a vertex and all its descendants. The number of distinct fringe subtrees in random trees has been studied by several authors, notably because of its connection to tree compaction algorithms. Here, we obtain a very precise result for binary search trees: it is shown that the number of distinct fringe subtrees in a binary search tree with n leaves is asymptotically equal to c(1)n/log n for a constant c(1) approximate to 2.4071298335, both in expectation and with high probability. This was previously shown to be a lower bound, our main contribution is to prove a matching upper bound. The method is quite general and can also be applied to similar problems for other tree models.

The spectral radius of a graph is the spectral radius of its adjacency matrix. A threshold graph is a simple graph whose vertices can be ordered as v_1, v_2, …, v_n, so that for each 2 ≤ i ≤ n, vertex v_i is either adjacent or nonadjacent simultaneously to all of v_1, v_2, …, v_i-1. Brualdi and Hoffman initially posed and then partially solved the extremal problem of finding the simple graphs with a given number of edges that have the maximum spectral radius. This problem was subsequently completely resolved by Rowlinson. Here, we deal with the similar problem of maximizing the spectral radius over the set of connected simple graphs with a given number of vertices and edges. As shown by Brualdi and Solheid, each such extremal graph is necessarily a threshold graph. We investigate the spectral radii of threshold graphs by relying on computations involving lazy walks. Furthermore, we obtain three lower bounds and one upper bound on the spectral radius of a given connected threshold graph.

A subdiagonal composition of a positive integer is a composition with the property that the ith part is less than or equal to i, and analogously a superdiagonal composition is a composition with the property that the ith part is greater than or equal to i. The generating functions for subdiagonal and superdiagonal compositions as well as for compositions with a larger class of lower and upper boundary conditions are obtained. The asymptotics estimates for the numbers in these various classes of compositions are found as the size n of the composition tends to infinity.