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 $\{1,2,\mathrm{.}\mathrm{.}\mathrm{.},n\}$ with cardinality $\ge 2$ and the set of partitions of $\{1,2,\mathrm{.}\mathrm{.}\mathrm{.},n+k-1\}$ into $k$ parts with cardinality $\ge 2$.