Hook Weighted Increasing Trees, Cayley Trees and Abel-Hurwitz Identities

TL;DR

This paper establishes a bijection between weighted increasing trees and Cayley trees, providing combinatorial proofs of hook summation formulas and connecting these to Abel-Hurwitz identities.

Contribution

It introduces a bijection solving a posed problem and links hook formulas to classical identities, with new combinatorial proofs and applications.

Findings

Established a bijection between weighted increasing trees and Cayley trees.

Provided two simple combinatorial proofs of the hook summation formula.

Connected hook summation formulas to Abel-Hurwitz identities and generalized theorems.

Abstract

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 weighted increasing trees and Cayley trees. We give such a bijection, providing an answer to the problem posed by F\'eray, Goulden and Lascoux as well as showing a combinatorial connection to the theory of tree volumes defined by Kelmans. In addition we give two simple proofs of the hook summation formula. As an application we describe how the hook summation formula gives a combinatorial proof of a generalization of Abel and Hurwitz' theorem, originally proven by Strehl.