The Kruskal-Katona theorem and a characterization of system signatures

We show how to determine if a given vector can be the signature of a system on a finite number of components and, if so, exhibit such a system in terms of its structure function. The method employs combinatorial results from the theory of (finite) simplicial complexes, and provides a full characterization of signature vectors using a theorem of Kruskal and Katona. We also show how the same approach can provide new combinatorial proofs of further results, e.g., that the signature vector of a system cannot have isolated zeroes. Last, we prove that a signature with all nonzero entries must be the uniform distribution.