(k, m)-Catalan numbers and hook length polynomials for plane trees

Motivated by a formula of A. Postnikov relating binary trees, we define the hook length polynomials for m-ary trees and plane forests, and show that these polynomials have a simple binomial expression. An integer value of this expression is C-k,C-m (n) = 1/mn+1 ((mn+1)k(n)), which we call the (k, m)-Catalan number. For proving the hook length formulas, we also introduce a combinatorial family, (k, m)-ary trees, which are counted by the (k, m)-Catalan numbers. (c) 2006 Elsevier Ltd. All rights reserved.