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

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