Nested sets, set partitions and Kirkman-Cayley dissection numbers

TL;DR

This paper provides a bijective proof of the Kirkman-Cayley formula for convex polygon dissections, linking nested sets and set partitions to establish the enumeration.

Contribution

It introduces an explicit bijection connecting nested sets and set partitions, offering a novel combinatorial proof of the Kirkman-Cayley formula.

Findings

Established a bijective correspondence between nested sets and set partitions.

Provided a combinatorial proof of the Kirkman-Cayley dissection formula.

Enhanced understanding of polygon dissection enumeration through combinatorics.

Abstract

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 \(\geq 2\).