Multivariate Fuss-Catalan numbers

TL;DR

This paper explores multivariate Fuss-Catalan numbers, extending classical Catalan numbers to higher dimensions, providing explicit formulas, combinatorial interpretations, and generalizations to p-dimensional arrays for counting p-ary trees.

Contribution

It introduces and analyzes higher-dimensional Fuss-Catalan numbers, offering explicit formulas, combinatorial descriptions, and generalizations to p-dimensional arrays.

Findings

Derived explicit formulas for multivariate Fuss-Catalan numbers.

Provided combinatorial interpretations in terms of trees and paths.

Extended the construction to p-dimensional arrays for p-ary trees.

Abstract

Catalan numbers $C(n)=\frac{1}{n+1}{2n\choose n}$ enumerate binary trees and Dyck paths. The distribution of paths with respect to their number $k$ of factors is given by ballot numbers $B(n,k)=\frac{n-k}{n+k}{n+k\choose n}$. 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 $C_3(n)=\frac 1 {2n+1} {3n\choose n}$, 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 $C_p(n)=\frac 1 {(p-1)n+1} {pn\choose n}$, the number of $p$-ary trees.