On Han's Hook Length Formulas for Trees
William Y.C. Chen, Oliver X.Q. Gao, and Peter L. Guo
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 associated with staircase labelings.
AMS Classification: 05A15, 05A19
Keywords: hook length formula, k-ary tree, bijection, staircase labeling