Sign patterns that require or allow particular refined inertias

The refined inertia ri(A) of a real n×n matrix A is the ordered 4-tuple (n+,n-,nz,2np) where n+ (resp. n-,nz,2np) is the number of positive (resp. negative, zero, nonzero pure imaginary) eigenvalues of A. Let Hn={(0,n,0,0),(0,n-2,0,2),(2,n-2,0,0)}. An n×n sign pattern Sn requires Hn if Hn={ri(A)|A has sign pattern Sn} and allows Hn if Hn∈{ri(A)|A has sign pattern Sn}. Sign patterns that require or allow Hn are investigated for small values of n and for patterns with all diagonal entries negative. Examples are given relating these concepts to Hopf bifurcation in dynamical systems.