  We explore how the combinatorial arrangement of prescribed zeros in a matrix
affects the possible eigenvalues that the matrix can obtain. We demonstrate
that there are inertially arbitrary patterns having a digraph with no 2-cycle,
unlike what happens for nonzero patterns. We develop a class of patterns that
are refined inertially arbitrary but not spectrally arbitrary, making use of
the property of a properly signed nest. We include a characterization of the
inertially arbitrary and refined inertially arbitrary patterns of order three,
as well as the patterns of order four with the least number of nonzero entries.
