(k,m)(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 mm-ary trees and plane forests, and show that these polynomials have a simple binomial expression. An integer value of this expression is Ck,m(n)=1mn+1(mn+1)kn, which we call the (k,m)(k,m)-Catalan number. For proving the hook length formulas, we also introduce a combinatorial family, (k,m)(k,m)-ary trees, which are counted by the (k,m)(k,m)-Catalan numbers.