Related papers: Hook length polynomials for plane forests of a cer…
We present a simple combinatorial proof of Postnikov's hook length formula for binary trees.
Motivated by a formula of A. Postnikov relating binary trees, we define the hook length polynomials for m-ary trees and plane forests, and show that these polynomials have a simple binomial expression. An integer value of this expression is…
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.
We introduce the hook length expansion technique and explain how to discover old and new hook length formulas for partitions and plane trees. The new hook length formulas for trees obtained by our method can be proved rather easily, whereas…
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…
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 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…
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 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…
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.
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…
Several hook summation formulae for binary trees have appeared recently in the literature. In this paper we present an analogous formula for unordered increasing trees of size r, which involves r parameters. The right-hand side can be…
Han recently discovered new hook length identities for binary trees. In this paper, we extend Han's identities to binomial families of trees. Moreover, we present a bijective proof of one of the identities for the family of ordered trees.
In this paper, we give a simple combinatorial explanation of a formula of A. Postnikov relating bicolored rooted trees to bicolored binary trees. We also present generalized formulas for the number of labeled k-ary trees, rooted labeled…
A nice factorization is given for the characteristic polynomials of intervals in some posets of leaf-labeled forests of rooted binary trees.
In this paper, we define two kinds of hook length for internal vertices of complete $m$-ary trees, and deduce their corresponding hook length formulas, which generalize the main results obtained by Du and Liu.
In a first part, we formalize the construction of combinatorial Hopf algebras from plactic-like monoids using polynomial realizations. Thank to this construction we reveal a lattice structure on those combinatorial Hopf algebras. As an…
We define, for an arbitrary partially ordered set, a multi-variable polynomial generalizing the hook polynomial.
In this paper, we take interest in finding applications for a hook-length formula recently proved in (Morales Pak Panova 2016). This formula can be applied to give a non trivial relation between alternating permutations and weighted Dyck…
A multiset hook length formula for integer partitions is established by using combinatorial manipulation. As special cases, we rederive three hook length formulas, two of them obtained by Nekrasov-Okounkov, the third one by Iqbal, Nazir,…