New hook length formulas for binary trees

— We find two new hook length formulas for binary trees. The particularity of our formulas is that the hook length h v appears as an exponent. Consider the set B (n) of all binary trees with n vertices. It is well-known that the cardinality of B (n) is equal to the Catalan number (see, e.g., [9, p.220]): (1) T ∈B(n) 1 = 1 n + 1 2n n. For each vertex v of a binary tree T ∈ B (n) the hook length of v, denoted by h v (T) or h v , is the number of descendants of v (including v). It is also well-known [4, p.67] that the number of ways to label the vertices of T with {1, 2,. .. , n}, such that the label of each vertex is less than that of its descendants, is equal to n! divided by the product of the h v 's (v ∈ T). On the other hand, each labeled binary tree with n vertices is in bijection with a permutation of order n [8, p.24], so that (2) T ∈B(n) n! v∈T 1 h v = n! The following hook length formula for binary trees (3) T ∈B(n) n! 2 n v∈T 1 + 1 h v = (n + 1) n−1 is due to Postnikov [6]. Further combinatorial proofs and extensions have been proposed by several authors [1, 2, 3, 5, 7]. In the present Note we obtain the following two new hook length formulas for binary trees. The particularity of our formulas is that the hook length h v appears as an exponent. Their proofs are based on the induction principle. It would be interesting to find simple bijective proofs.