Pattern-avoiding (0,1)-matrices and bases of permutation matrices

We investigate pattern-avoiding n × n (0, 1)-matrices with emphasis on patterns of length 3: pqr-avoiding where {p, q, r} ⊆ {1, 2, . . . , n}. We show that all such maximal (0, 1)-matrices contain the same number of 1’s, and their structure is determined. We then show that the set of pqr-avoiding n × n permutation matrices span the linear space of dimension (n − 1)2 + 1 generated by the n × n permutation matrices and determine a corresponding basis for each p, q, r.