A combinatorial generalization of the Durfee square

Abstract

The Durfee square of a partition λ, D(λ), is defined as the largest square contained in the shape of λ.It was proved in [2] (cf. also [3]) that the size of D(λ), d(λ), was related to the perfection of a certain module M_λ , an algebro-geometric object (cf. also [1], [4], [5]). The goal of this note is to propose a generalization of the notion of Durfee square to the case of a pair (α, β) of partitions.