#### 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
##### Milestones
Received: 2 September 2014
Revised: 6 February 2015
Accepted: 6 February 2015
Published: 2 March 2016

Communicated by Kenneth S. Berenhaut
##### Authors
 Alison Schuetz Department of Mathematics Hood College 401 Rosemont Avenue Frederick, MD 21701 United States Gwyn Whieldon Department of Mathematics Hood College 401 Rosemont Avenue Frederick, MD 21701 United States