Tree hook length formulae, Feynman rules and B-series

TL;DR

This paper unifies and generalizes various results on weighted generating functions of trees, connecting combinatorics, differential equations, and quantum field theory to derive new formulas and examples.

Contribution

It unifies three independent results on tree generating functions and introduces new hook length formulas using differential equations and quantum field theory insights.

Findings

New hook length formulas for trees derived from differential equations

Unified framework for results from different communities

Quantum field theory examples with combinatorial properties

Abstract

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 the same result concerning these series. We unify these results with a common generalization. Next we use the insights of one community on the problems of another in two different ways. Namely, we use the differential equation perspective to find a number of new interesting hook length formulae for trees, and we use the body of examples developed by the combinatorial community to give quantum field theory toy examples with nice properties.