Partitions versus sets : a case of duality

TL;DR

This paper generalizes a duality framework between decompositions and their dual objects, providing a simpler proof and deeper insight into the conditions for duality in combinatorial structures.

Contribution

It extends Amini et al.'s framework and offers an accessible proof, clarifying the reasons behind the existence of dual objects for certain decompositions.

Findings

Unified duality theorem for decompositions and dual objects

Simplified proof enhances understanding of duality conditions

Generalization broadens applicability of the duality framework

Abstract

In a recent paper, Amini et al. introduce a general framework to prove duality theorems between special decompositions and their dual combinatorial object. They thus unify all known ad-hoc proofs in one single theorem. While this unification process is definitely good, their main theorem remains quite technical and does not give a real insight of why some decompositions admit dual objects and why others do not. The goal of this paper is both to generalise a little this framework and to give an enlightening simple proof of its central theorem.