@article{AVAL20084660,
    Abstract = {Catalan numbers C(n)=1/(n+1)2nn 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+kn. 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)3nn, which may be called order-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)pnn, the number of p-ary trees.},
    Author = {Aval, Jean-Christophe},
    File = {Multivariate Fuss–Catalan numbers - 1-s2.0-S0012365X07007108-main.pdf},
    ISSN = {0012-365X},
    Journal = {Discrete Mathematics},
    Keywords = {Catalan numbers, Statistics on paths and trees},
    Number = {20},
    Pages = {4660-4669},
    Title = {Multivariate Fuss--Catalan numbers},
    URL = {https://www.sciencedirect.com/science/article/pii/S0012365X07007108},
    Volume = {308},
    Year = {2008},
    bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/S0012365X07007108},
    bdsk-url-2 = {https://doi.org/10.1016/j.disc.2007.08.100},
    date-added = {2022-10-30 09:21:53 +0100},
    date-modified = {2022-10-30 09:21:53 +0100},
    doi = {10.1016/j.disc.2007.08.100}
}

@article{AVAL20084660, Abstract = {Catalan numbers C(n)=1/(n+1)2nn 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+kn. 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)3nn, which may be called order-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)pnn, the number of p-ary trees.}, Author = {Aval, Jean-Christophe}, File = {Multivariate Fuss–Catalan numbers - 1-s2.0-S0012365X07007108-main.pdf}, ISSN = {0012-365X}, Journal = {Discrete Mathematics}, Keywords = {Catalan numbers, Statistics on paths and trees}, Number = {20}, Pages = {4660-4669}, Title = {Multivariate Fuss--Catalan numbers}, URL = {https://www.sciencedirect.com/science/article/pii/S0012365X07007108}, Volume = {308}, Year = {2008}, bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/S0012365X07007108}, bdsk-url-2 = {https://doi.org/10.1016/j.disc.2007.08.100}, date-added = {2022-10-30 09:21:53 +0100}, date-modified = {2022-10-30 09:21:53 +0100}, doi = {10.1016/j.disc.2007.08.100} }