Calculating the cardinality of some classes of binary matrices using bitwise operations — A polynomial algorithm

Abstract

Let [Formula: see text] be the set of all [Formula: see text] binary matrices with exactly [Formula: see text] units in each row and each column, [Formula: see text]. A matrix [Formula: see text] will be called primitive, if there is no [Formula: see text] submatrix of [Formula: see text] that belongs to the set [Formula: see text], [Formula: see text]. The article describes a polynomial algorithm, which works in time [Formula: see text] for verifying whether a [Formula: see text]-matrix is primitive. The work applies this algorithm for finding all primitive [Formula: see text]-matrices which rows and columns are in lexicographically nondecreasing order (semi-canonical binary matrices) for some integers [Formula: see text] and [Formula: see text].