Polygonal dissections and reversions of series

Abstract

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 $C_k^{\left(d\right)}$ count enumerations of dissections of polygons $P_{k\left(d-1\right)+2}$ into $\left(d+1\right)$-gons. In this paper, we provide a formula enumerating polygonal dissections of $\left(n+2\right)$-gons, classified by partitions $\lambda$ of $\left[n\right]$. 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.