Related papers: A multivariate hook formula for labelled trees
A number of hook formulas and hook summation formulas have previously appeared, involving various classes of trees. One of these classes of trees is rooted trees with labelled vertices, in which the labels increase along every chain from…
Recently F\'eray, Goulden and Lascoux gave a proof of a new hook summation formula for unordered increasing trees by means of a generalization of the Pr\"ufer code for labelled trees and posed the problem of finding a bijection between…
In this short note we discuss recent results on hook length formulas of trees unifying some earlier results, and explain hook length formulas naturally associated to families of increasingly labelled trees.
Recently, Han discovered two formulas involving binary trees which have the interestig property that hooklengths appear as exponents. The purpose of this note is to give a probabilistic proof of one of Han's formulas. Yang has generalized…
We present a simple combinatorial proof of Postnikov's hook length formula for binary trees.
Recently, Han obtained two hook length formulas for binary trees and asked for combinatorial proofs. One of Han's formulas has been generalized to k-ary trees by Yang. Sagan has found a probabilistic proof of Yang's extension. We give…
We discover another one-parameter generalization of Postnikov's hook length formula for binary trees. The particularity of our formula is that the hook length $h_v$ appears as an exponent. As an application, we derive another simple hook…
We find two new hook length formulas for binary trees. The particularity of our formulas is that the hook length $h_v$ appears as an exponent.
In this work we introduce and study various generalizations of the notion of increasingly labelled trees, where the label of a child node is always larger than the label of its parent node, to multilabelled tree families, where the nodes in…
Recently Han obtained a general formula for the weight function corresponding to the expansion of a generating function in terms of hook lengths of binary trees. In this paper, we present formulas for k-ary trees, plane trees, plane…
A proper vertex of a rooted tree with totally ordered vertices is a vertex that is less than all its proper descendants. We count several kinds of labeled rooted trees and forests by the number of proper vertices. Our results are all…
We provide a short combinatorial proof of Cayley's formula by means of a bijective map to an outcome space of an urn-drawing problem. Furthermore we introduce an algebraic structure on the set of labeled trees, which provides a more…
We give a short proof of Cayley's tree formula for counting the number of different labeled trees on $n$ vertices. The following nonlinear recursive relation for the number of labeled trees on $n$ vertices is deduced from a combinatorial…
We consider weighted generating functions of trees where the weights are products of functions of the sizes of the subtrees. This work begins with the observation that three different communities, largely independently, found substantially…
Cayley's formula is a fundamental result in combinatorics that counts the number of labeled trees on n vertices. While existing proofs use approaches such as Prufer sequences and the Matrix-Tree Theorem, we give a combinatorial proof that…
We prove a new formula for the generating function of multitype Cayley trees counted according to their degree distribution. Using this formula we recover and extend several enumerative results about trees. In particular, we extend some…
This paper introduces a new combinatorial framework for modeling the growth of binary trees through a discrete evolution process that incorporates a growing rule and an extinction rule. Building upon the theory of increasingly labeled…
We consider the counting problem of the number of \textit{leaf-labeled increasing trees}, where internal nodes may have an arbitrary number of descendants. The set of all such trees is a discrete representation of the genealogies obtained…
The original motivation for study for hook length polynomials was to find a combinatorial proof for a hook length formula for binary trees given by Postnikov, as well as a proof for a hook length polynomial formula conjectured by Lascoux.…
We prove a weighted generalization of the formula for the number of plane vertex-labeled trees.