Sets of refined inertias of zero-nonzero patterns

A set Hn⁎ of refined inertias for zero-nonzero patterns is introduced that is analogous to the set Hn previously considered for sign patterns. For n=3 and 4, a complete characterization of irreducible zero-nonzero patterns that allow or require Hn⁎ is given, and each zero-nonzero pattern that allows Hn⁎ has a signing that allows Hn. In contrast, for n≥5 a family of irreducible zero-nonzero patterns is given that allows Hn⁎ but for which no one signing allows Hn.