Polygonal dissections and reversions of series

Schuetz, Alison; Whieldon, Gwyn

doi:10.2140/involve.2016.9.223

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Polygonal dissections and reversions of series

Alison Schuetz and Gwyn Whieldon

Vol. 9 (2016), No. 2, 223–236

DOI: 10.2140/involve.2016.9.223

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 $C_{k}^{(d)}$ 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 $λ$ of $[n]$ . We connect these counts $a_{λ}$ 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

Vol. 9, No. 2, 2016