  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 $\Delta$, we construct a complex $\Gamma \supseteq
\Delta$ of the same dimension such that both $\Gamma$ and the relative complex
$(\Gamma,\Delta)$ 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 $\Delta$ it is not always possible to find a
complex $\Gamma$ such that both $\Gamma$ and $(\Gamma,\Delta)$ are
Cohen-Macaulay. We characterize when this is possible, and we show that the
construction of such a $\Gamma$ 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.
