Partition and Cohen–Macaulay extenders

If a pure simplicial complex is partitionable, then its h-vector has a combinatorial interpretation in terms of any partitioning of the complex. Given a non-partitionable complex Δ, we construct a complex Γ⊇Δ of the same dimension such that both Γ and the relative complex (Γ,Δ) are partitionable. This allows us to rewrite the h-vector of any pure simplicial complex as the difference of two h-vectors of partitionable complexes, giving an analogous interpretation of the h-vector of a non-partitionable complex. By contrast, for a given complex Δ it is not always possible to find a complex Γ such that both Γ and (Γ,Δ) are Cohen–Macaulay. We characterize when this is possible, and we show that the construction of such a Γ in this case is remarkably straightforward. We end with a note on a similar notion for shellability and a connection to Simon's conjecture on extendable shellability for uniform matroids.