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The notions of $k$-separability and $k$-producibility are useful andexpressive tools for the characterization of entanglement in multipartitequantum systems, when a more detailed analysis would be infeasible or simplyneedless. In this work we reveal a partial duality between them, which is validalso for their correlation counterparts. This duality can be seen from a muchwider perspective, when we consider the entanglement and correlation propertieswhich are invariant under the permutations of the subsystems. These propertiesare labeled by Young diagrams, which we endow with a refinement-like partialorder, to build up their classification scheme. This general treatment revealsa new property, which we call $k$-stretchability, being sensitive in a balancedway to both the maximal size of correlated (or entangled) subsystems and theminimal number of subsystems uncorrelated with (or separable from) one another.

k-stretchability of entanglement, and the duality of k-separability and k-producibility