Related papers: Hook Length Formulas for Trees by Han's Expansion
We give factorizations for weighted spanning tree enumerators of Cartesian products of complete graphs, keeping track of fine weights related to degree sequences and edge directions. Our methods combine Kirchhoff's Matrix-Tree Theorem with…
We study the average height of random trees generated by leaf-centric binary tree sources as introduced by Zhang, Yang and Kieffer. A leaf-centric binary tree source induces for every $n \geq 2$ a probability distribution on the set of…
We introduce weighted regular tree grammars with storage as combination of (a) regular tree grammars with storage and (b) weighted tree automata over multioperator monoids. Each weighted regular tree grammar with storage generates a…
Maxmin trees are labeled trees with the property that each vertex is either a local maximum or a local minimum. Such trees were originally introduced by Postnikov, who gave a formula to count them and different combinatorial interpretations…
We introduce a generalization of Smirnov words in the context of labeled binary trees, which we call Smirnov trees. We study the generating function for ascent-descent statistics on Smirnov trees and establish that it is $e$-positive, which…
Michael Somos conjectured a relation between Hankel determinants whose entries $\frac 1{2n+1}\binom{3n}n$ count ternary trees and the number of certain plane partitions and alternating sign matrices. Tamm evaluated these determinants by…
In this work, we present a series of bijections that reveal the deep connections between the concepts of tree records, the girth of a connected endofunction, and the genesis sequence, the first sequence in the OEIS. We use these results to…
There are several interrelated notions of discrete curvature on graphs. Many approaches utilize the optimal transportation metric on its probability simplex or the distance matrix of the graph. In this survey article, we compute formulas…
A relaxed $k$-ary tree is an ordered directed acyclic graph with a unique source and sink in which every node has out-degree $k$. These objects arise in the compression of trees in which some repeated subtrees are factored and repeated…
This paper considers linear functions constructed on two different weighted branching processes and provides explicit bounds for their Kantorovich-Rubinstein distance in terms of couplings of their corresponding generic branching vectors.…
It is known that the number of minimal factorizations of the long cycle in the symmetric group into a product of $k$ cycles of given lengths has a very simple formula: it is $n^{k-1}$ where $n$ is the rank of the underlying symmetric group…
In this paper, we introduce a new representation for ordered trees, the weight sequence representation. We then use this to construct new representations for both rooted trees and free trees, namely the canonical weight sequence…
Hypertree decompositions (HDs), as well as the more powerful generalized hypertree decompositions (GHDs), and the yet more general fractional hypertree decompositions (FHDs) are hypergraph decomposition methods successfully used for…
Two-side estimates for two-weighted discrete Hardy-type operators on a tree are obtained. For general weights we prove the discrete analogue of Evans - Harris - Pick theorem (it is a quite simple consequence from their result). It gives the…
We prove limit theorems for sums of functions of subtrees of binary search trees and random recursive trees. In particular, we give simple new proofs of the fact that the number of fringe trees of size $ k=k_n $ in the binary search tree…
We establish limit theorems that describe the asymptotic local and global geometric behaviour of random enriched trees considered up to symmetry. We apply these general results to random unlabelled weighted rooted graphs and uniform random…
We study a generating function for the sum over fatgraphs with specified valences of vertices and faces, inversely weighted by the order of their symmetry group. A compact expression is found for general (i.e. non necessarily connected)…
In light of the grammar given by Ji for the $(\alpha,\beta)$-Eulerian polynomials introduced by Carlitz and Scoville, we provide a labeling scheme for increasing binary trees. In this setting, we obtain a combinatorial interpretation of the…
The $k$-expansion of a graph $G$ is the $k$-uniform hypergraph obtained from $G$ by adding $k-2$ new vertices to every edge. We determine, for all $k > d \geq 1$, asymptotically optimal $d$-degree conditions that ensure the existence of all…
We give a simple formula for the number of hypertrees with $k$ hyperedges of given sizes and $n+1$ labelled vertices with prescribed degrees. A slight generalization of this formula counts labelled bipartite trees with prescribed degrees in…