(k,m)-Catalan Numbers and Hook Length Polynomials for Plane Trees

Details (k,m)-Catalan Numbers and Hook Length Polynomials for Plane Trees (k,m)-Catalan Numbers and Hook Length Polynomials for Plane Trees

2005-01-11

arxiv.org/abs/math/0501147v2

Mathematics

Combinatorics

11 pages

Scientific paper

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)=\frac{1}{mn+1}{(mn+1)k \choose 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.

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