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

Authors:
Rosena R. X. Du;Fu Liu
Affiliations:
Department of Mathematics, East China Normal University, Shanghai 200062, PR China and Center for Combinatorics, LPMC, Nankai University, Tianjin 300071, PR China;Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Venue:
European Journal of Combinatorics
Year:
2007

Citing 1
Cited 3

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Abstract

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)=1mn+1(mn+1)kn, 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.