On Han's Hook Length Formulas for Trees

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.

A multivariate hook formula for labelled trees

“…• Another direction consists in replacing in summation formula (2) (or in the generalized version (3)) binary trees by other families of trees. Formulae of this kind for plane forests or m-ary trees have been given in several papers [13,34,33,10]; • Finally, formulae (1) and (2) admit a number of higher level interpretations.…”

“…This construction can be found in paper [19] (see in particular Theorem 3.2 and Proposition 4.1, which corresponds to the properties above). Note that it is well-known [14, Lemma 3.9] that the coefficients in the right-hand side of (10) do not depend on n (because |λ| = |µ| + |ν|).…”

Hook Length Formulas for Trees by Han's Expansion

“…The above formula (2.2) was derived by Han [8] for k = 2. Yang [16] showed that (2.2) holds for general k. Probabilistic and combinatorial proofs of (2.2) have been given by Sagan [13], and Chen, Gao and Guo [1], respectively.…”