The Catalan numbers
were first studied by Euler, in the context of enumerating triangulations of polygons
.
Among the many generalizations of this sequence, the Fuss–Catalan numbers
count enumerations of
dissections of polygons
into
-gons.
In this paper, we provide a formula enumerating polygonal dissections of
-gons, classified
by partitions
of . We connect
these counts
to reverse series arising from iterated polynomials. Generalizing this
further, we show that the coefficients of the reverse series of polynomials
enumerate colored polygonal dissections.
Keywords
Catalan, Fuss–Catalan, series reversion, Lagrange
inversion, polygon partitions