Symmetric matrix, Skew-symmetric matrix, Permanent rank, Principal permanent rank characteristic sequenc, generalized cycle, matching, minor
The principal permanent rank characteristic sequence is a binary sequence r_0 r_1 · · · r_n, where r_k = 1 if there exists a principal square submatrix of size k with nonzero permanent and r_k = 0 otherwise, and r_0 = 1 if there is a zero diagonal entry. A characterization is provided for all principal permanent rank sequences obtainable by the family of nonnegative matrices as well as the family of nonnegative symmetric matrices. Constructions for all realizable sequences are provided. Results for skew-symmetric matrices are also included.
Hassani Monfared, Keivan; Horn, Paul; Kenter, Franklin H. J.; Nowak, Kathleen; Sinkovic, John; and Tobin, Josh.
"On the Principal Permanent Rank Characteristic Sequences of Graphs and Digraphs",
Electronic Journal of Linear Algebra,
Volume 31, pp. 187-199.