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

Abstract

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.