Hook Length Formulas for Trees by Han's Expansion

TL;DR

This paper extends Han's hook length formulas from binary trees to various other tree structures, providing unified generating functions and broadening the combinatorial understanding of hook length formulas.

Contribution

It introduces new hook length formulas for k-ary trees, plane trees, forests, and unifies existing formulas through generating functions.

Findings

Formulas for k-ary trees, plane trees, and forests

Unified generating functions for hook length formulas

Connections to previous formulas by Du, Liu, Han, Gessel, Seo, and Postnikov

Abstract

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 forests, labeled trees and forests. We also find appropriate generating functions which lead to unifications of the hook length formulas due to Du and Liu, Han, Gessel and Seo, and Postnikov.