Vol. 9, No. 2, 2016

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Polygonal dissections and reversions of series

Alison Schuetz and Gwyn Whieldon

Vol. 9 (2016), No. 2, 223–236
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.

Keywords
Catalan, Fuss–Catalan, series reversion, Lagrange inversion, polygon partitions
Mathematical Subject Classification 2010
Primary: 05A15, 05E99