  We present an affirmative answer to Stanley's zrank problem, namely, the
zrank and rank are equal for any skew partition. We show that certain classes
of restricted Cauchy matrices are nonsingular and furthermore, the signs depend
on the number of zero entries. Similar to notion of the jrank of a skew
partition, we give a characterization of the rank in terms of the Giambelli
type matrices of the corresponding skew Schur functions. We also show that the
sign of the determinant of a factorial Cauchy matrix is uniquely determined by
the number of its zero entries, which implies the nonsingularity of the inverse
binomial coefficient matrix.
