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

The Catalan numbers Ck were first studied by Euler, in the context of enumerating triangulations of polygons Pk+2. Among the many generalizations of this sequence, the Fuss–Catalan numbers Ck(d) count enumerations of dissections of polygons Pk(d1)+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 aλ to reverse series arising from iterated polynomials. Generalizing this further, we show that the coefficients of the reverse series of polynomials x = z j=0bjzj+1 enumerate colored polygonal dissections.

Catalan, Fuss–Catalan, series reversion, Lagrange inversion, polygon partitions
Mathematical Subject Classification 2010
Primary: 05A15, 05E99
Received: 2 September 2014
Revised: 6 February 2015
Accepted: 6 February 2015
Published: 2 March 2016

Communicated by Kenneth S. Berenhaut
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