Multivariate Fuss-Catalan numbers
| dc.creator | Aval, Jean-Christophe | |
| dc.date | 2007-11-06 | |
| dc.date.accessioned | 2026-07-07T10:14:22Z | |
| dc.date.available | 2026-07-07T10:14:22Z | |
| dc.description | 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. | |
| dc.identifier | https://arxiv.org/abs/0711.0906 | |
| dc.identifier | http://arxiv.org/abs/0711.0906 | |
| dc.identifier | Discrete Mathematics 308 (2008) 4660-4669 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/172859 | |
| dc.subject | Combinatorics | |
| dc.title | Multivariate Fuss-Catalan numbers | |
| dc.type | text |