Refined inertias of positive and hollow positive patterns

We investigate refined inertias of positive patterns and patterns that have each off-diagonal entry positive and each diagonal entry zero, i.e., hollow positive patterns. For positive patterns, we prove that every refined inertia with can be realized. For hollow positive patterns, we prove that every refined inertia with and can be realized. To illustrate these results, we construct matrix realizations using circulant matrices and bordered circulants. For both patterns of order , we show that as , the fraction of possible refined inertias realized by circulants approaches 1/4 for odd and 3/4 for even.