  The notions of $k$-separability and $k$-producibility are useful and
expressive tools for the characterization of entanglement in multipartite
quantum systems, when a more detailed analysis would be infeasible or simply
needless. In this work we reveal a partial duality between them, which is valid
also for their correlation counterparts. This duality can be seen from a much
wider perspective, when we consider the entanglement and correlation properties
which are invariant under the permutations of the subsystems. These properties
are labeled by Young diagrams, which we endow with a refinement-like partial
order, to build up their classification scheme. This general treatment reveals
a new property, which we call $k$-stretchability, being sensitive in a balanced
way to both the maximal size of correlated (or entangled) subsystems and the
minimal number of subsystems uncorrelated with (or separable from) one another.
