Polygonal Dissections and Reversions of Series

The Catalan numbers \(C_k\) were first studied by Euler, in the context of enumerating triangulations of polygons \(P_{k+2}\). Among the many generalizations of this sequence, the Fuss-Catalan numbers \(C^{(d)}_k\) count enumerations of dissections of polygons \(P_{k(d-1)+2}\) into \((d+1)\)-gons. In this paper, we provide a formula enumerating polygonal dissections of \((n+2)\)-gons, classified by partitions \(\lambda\) of \([n]\). We connect these counts \(a_{\lambda}\) to reverse series arising from iterated polynomials. Generalizing this further, we show that the coefficients of the reverse series of polynomials \(x=z-\sum_{j=0}^{\infty} b_j z^{j+1}\) enumerate colored polygonal dissections.