A multivariate hook formula for labelled trees

TL;DR

This paper introduces a new multivariate hook formula for unordered increasing trees, generalizing existing formulas for binary trees, and provides multiple proofs including one based on symmetric group representation theory.

Contribution

It presents an analogous hook formula for unordered increasing trees involving multiple parameters, along with three different proofs, one of which uses symmetric group representation theory.

Findings

Derived a new hook formula for unordered increasing trees

Connected the formula to Cayley's enumeration of trees by vertex degree

Provided three proofs, including one using representation theory

Abstract

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 written nicely as a product of linear factors. We study two specializations of this new formula, including Cayley's enumeration of trees with respect to vertex degree. We give three proofs of the hook formula. One of these proofs arises somewhat indirectly, from representation theory of the symmetric groups, and in particular uses Kerov's character polynomials. The other proofs are more direct, and of independent interest.