On Han's Hook Length Formulas for Trees

TL;DR

This paper provides combinatorial proofs for Han's hook length formulas for binary and k-ary trees, using bijections based on staircase labelings, advancing understanding of tree enumeration formulas.

Contribution

It offers the first combinatorial proofs of Han's hook length formulas for binary and k-ary trees, complementing existing probabilistic proofs.

Findings

Combinatorial proofs for Han's formulas for binary trees

Extension of Han's formulas to k-ary trees

Bijections based on staircase labelings for tree structures

Abstract

Recently, Han obtained two hook length formulas for binary trees and asked for combinatorial proofs. One of Han's formulas has been generalized to k-ary trees by Yang. Sagan has found a probabilistic proof of Yang's extension. We give combinatorial proofs of Yang's formula for k-ary trees and the other formula of Han for binary trees. Our bijections are based on the structure of k-ary trees with staircase labelings.