Multivariate Fuss–Catalan numbers

Abstract

Catalan numbers C ( n ) = 1 / ( n + 1 ) 2 n 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 B 3 ( n , k , l ) that give a 2-parameter distribution of C 3 ( n ) = 1 / ( 2 n + 1 ) 3 n n , which may be called order-3 Fuss–Catalan numbers, and enumerate ternary trees. The aim of this paper is a study of these integers B 3 ( 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 C p ( n ) = 1 / ( ( p - 1 ) n + 1 ) pn n , the number of p-ary trees.