Joel Persson: Generalisations of Interpretations of Fuss-Catalan numbers

Abstract:

The paper looks at the generalisation of the Catalan numbers \(a_k(n)\) defined by \(a_k(0) = 1 \) and \(a_k(n+1) = P \, j≥0,k|j a_k(j)a_k(n−j) \)introduced by Per Alexandersson, Samuel Asefa Fufa, Frether Getachew, and Dun Qiu in 2022. We first look at combinatorial interpretations of the Fuss-Catalan numbers then we generalise these into interpretations of ak(n). Finally we give a combinatorial proof of the identity \(a_k(n) = A_m(k + 1, j + 1)\) where \(A_m(p, r)\) are the Raney-numbers.