Nested sets, set partitions and Kirkman-Cayley dissection numbers

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 \(\geq 2\).