Discovering hook length formulas by expansion technique

TL;DR

This paper introduces a new expansion technique for discovering hook length formulas for partitions and plane trees, providing a computational tool and new formulas, some of which remain conjectural.

Contribution

It presents the hook length expansion technique, a Maple package HookExp, and new hook length formulas for trees and partitions, including open conjectures.

Findings

New hook length formulas for trees and partitions

A Maple package HookExp for computation

Some formulas for partitions remain conjectural

Abstract

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 those for partitions are much more difficult and some of them still remain open conjectures. We also develop a Maple package HookExp for computing the hook length expansion. The paper can be seen as a collection of hook length formulas for partitons and plane trees. All examples are illustrated by HookExp and, for many easy cases, expained by well-known combinatorial arguments.