On Postnikov’s hook length formula for binary trees

Abstract

We present a simple combinatorial proof of Postnikov's hook length formula for binary trees.

Hook Length Polynomials for Plane Forests of a Certain Type

“…It's not hard to see that G r is connected. (1) and S (2) in V r , C F r, k, S (1) = C F r, k, S (2) .…”

“…We call two partitions S (1) and S (2) adjacent if there exists i ∈ [n − 1], such that we can obtain S (1) by swapping i and i + 1 in S (2) .…”

“…We define a map ψ from C F r, k, S (1) to C F r, k, S (2) . For any colored labelled forest F ∈ C F r, k, S (1) , let v 1 be the vertex with label i and v 2 be the vertex with label i + 1:…”

“…We construct a graph G r with vertex set V r and S (1) , S (2) forming an edge in G r if and only if they are adjacent. It's not hard to see that G r is connected.…”

“…where the sum is over all complete binary trees with n internal vertices, the product is over all internal vertices of T , and h v is the "hook length" of v in T , namely, the number of internal vertices in the subtree of T rooted at v. Chen and Yang [2] and Seo [9] both gave direct bijective proofs for (1.1). Moreover, based on (1.1), Lascoux replaced 1 with x and conjectured a hook length polynomial formula for binary trees:…”