A combinatorial generalization of the Durfee square

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 Mλ, 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. More precisely, given two partitions α and β s.t. the last row of β is shorter then the first row of α, we define in Section 2, a partition D(α, β) (not necessarily a square), which we call “generalized Durfee partition of α w.r.t. β”. D(α, β) is also related to algebro-geometric problems, as it is indicated for instance by the fact (proven in Section 3) that D(α, β) allows us to construct Lascoux’s rectification of α and β (cf. [8]). We conjecture that D(α, β) might encode important information on the minimal free resolution of significant classes of modules (see, for instance