Multivariate Fuss-Catalan numbers

Catalan numbers C(n) = 1 n+1 2n n enumerate binary trees and Dyck paths. The distribution of paths with respect to their number k of factors is given by ballot numbers B(n, k) = n-k n+k n+k n . These integers are known to satisfy simple recurrence, which may be visualised in a "Catalan triangle", a lower-triangular two-dimensional array. It is surprising that the extension of this construction to 3 dimensions generates integers B3(n, k, l) that give a 2-parameter distribution of C3(n) = 1 2n+1 3n n , which may be called oder-3 Fuss-Catalan numbers, and enumerate ternary trees. The aim of this paper is a study of these integers B3(n, k, l). We obtain an explicit formula and a description in terms of trees and paths. Finally, we extend our construction to p-dimensional arrays, and in this case we obtain a (p - 1)-parameter distribution of Cp(n) = 1 (p-1)n+1 pn n , the number of p-ary trees.

• Type: Journal
• Year: 2008
• Where: DM
• Authors: Jean-Christophe Aval
• Keywords: Binary Trees, DM 2008, Dyck Paths, Lower-triangular Two-dimensional Array