Nested sets, set partitions and Kirkman–Cayley dissection numbers

In this paper we show 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 kk subsets of {1,2,…,n}{1,2,…,n} with cardinality ≥2≥2 and the set of partitions of {1,2,…,n+k−1}{1,2,…,n+k−1} into kk blocks with cardinality ≥2≥2. A bijection between these two sets can be obtained from Péter L. Erdős and L.A. Székely result in Erdős and Székely (1989); to make this paper self contained we describe another explicit bijection that is a variant of their bijection.