The principal rank characteristic sequence of a real symmetric matrix

Given a vector u∈R2n, the principal minor assignment problem asks when is there an n×n matrix having its 2n principal minors given by u. This paper explores the following related problem. Given a sequence r0r1⋯rn of 0s and 1s, does there exist an n×n real symmetric matrix that has a principal submatrix of rank k if and only if rk=1, for all 0⩽k⩽n? Certain conditions are shown to be necessary in order for this question to have an affirmative answer. Several families of matrices are constructed to attain certain classes of sequences. The problem is solved completely for n⩽6, and for 7⩽n⩽10 in the case of sequences beginning with 010.