(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 , m ( n ) = 1 m n + 1 ( m n + 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.