Herbert J. Ryser

Combinatorial Properties of Matrices of Zeros and Ones

This paper is concerned with a matrix A of m rows and n columns, all of whose entries are 0’s and 1’s. Let the sum of row i of A be denoted by r i (i = 1, ... , m) and let the sum of column i of A be denoted by S i (i = 1, ... ,n).

Matrices of zeros and ones

Let A be a. matrix of m rows and n columns and let the entries of A be the integers 0 and 1. We call such a matrix a (0, 1) -matrix of size m by n. The 2 (0, 1)-matrices of size m by n play a fundamental role in a wide variety of combinatorial investigations. One of the chief reasons for this is the following. Let X be a set of n elements Xi, X2, • • • , xn and let Xi, X2, • • • , Xm be m subsets of X. Let dij= 1 if Xj is a member of Xi and let 0,7 = 0 if Xj is not a member of Xi. The a»/s yield a (0, 1)-matrix 4 = [a»y] of size w by w called the incidence matrix for the subsets Xi, X2, • • • , Xm of X. The l s in row i oî A specify the elements that belong to set Xi and the l s in column j of A specify the sets that contain element x3-. The matrix A characterizes the m subsets Xi, X2, • • • , Xm of the set X. Let A be a (0, 1)-matrix of size m by n. het the sum of row i of A be denoted by r» and let the sum of column j of 4̂ be denoted by Sj. We call

Multiplicities and minimal widths for (0,1)-matrices.

Abstract : The present analysis continues the study of alpha-width of a (0,1)- matrix. The principal new result is a simple construction that produces a matrix having the property that its alphawidths are minimal for all alphas. (Author)

Traces, term ranks, widths and heights

The notions of widths and heights of (0 1)-matrices are discussed in the general setting of known results concerning traces and term ranks. Proofs are omitted throughout.