  In this paper we show a a proof by explicit bijections of the famous
Kirkman-Cayley formula for the number of dissections of a convex polygon. Our
starting point is the bijective correspondence between the set of nested sets
made by \(k\) subsets of \(\{1,2,...,n\}\) with cardinality \(\geq 2\) and the
set of partitions of \(\{1,2,...,n+k-1\}\) into \(k\) parts with cardinality
\(\geq 2\).
