H.J. Ryser’s research while affiliated with The Ohio State University and other places

: The study of alpha-widths of (0, 1)-matrices (AD-274 181) continued, the emphasis being on those special classes of b by v (0, 1)-matrices having k 1's per row and 4 1's per column. It is assumed throughout that the class parameters b, v, k, r satisfy the inequality (b-r)(v-k-1) less than or equal to v - 1. Such a class has special combinatorial interest. For example, complements of finite projective planes and of Steiner triple systems have parameters satisfying this inequality. Several theorems are proved concerning the width sequence for a matrix in such a class. Insofar as possible, these results are used to obtain information concerning the maximal width sequence for the class. Perhaps the major general result established is that jumps in the width sequence for a matrix in the class, or in the maximal width sequence for the class, are either 1 or 2.

Let A be an m by n (0, 1)-matrix, and suppose that E* is an m by epsilon submatrix of A having the property that each row of E* contains at least alpha 1's. The epsilon columns of E* are said to form an alpha-set of representatives for A. Let epsilon(alpha) be the minimal number of columns of A that form an alpha-set of representatives. The integer epsilon(alpha) is called the alpha-width of A. If A has alpha-width epsilon(alpha), select an m by epsilon(alpha) submatrix E* of A having the property that the number delta(alpha) of rows of E* containing exactly alpha 1'S is as small as possible. The integer delta(alpha) is called the alpha-height of A.

In a previous paper (1) the notion of the α-width ∈ A (α) of a (0, 1)-matrix A was introduced, and a formula for the minimal α-width taken over the class of all (0, 1)-matrices having the same row and column sums as A , was obtained. The main tool in arriving at this formula was a block decomposition theorem (1, Theorem 2.1; repeated below as Theorem 2.1) that established the existence, in the class generated by A , of certain matrices having a simple block structure. The block decomposition theorem does not itself directly involve the notion of minimal α-width, but rather centres around a related class concept, that of multiplicity. We review both of these notions in § 2, together with some other pertinent definitions and results.

: Let A be an m by n (0, 1)-matrix, and suppose that E* is an m by epsilon submatrix of A having the property that each row of E* contains at least alpha 1's. The epsilon columns of E* are said to form an alpha-set of representatives for A. Let epsilon(alpha) be the minimal number of columns of A that form an alpha-set of representatives. The integer epsilon(alpha) is called the alpha-width of A. If A has alpha-width epsilon(alpha), select an m by epsilon(alpha) submatrix E* of A having the property that the number delta(alpha) of rows of E* containing exactly alpha 1'S is as small as possible. The integer delta(alpha) is called the alpha-height of A.

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.

This paper continues the study appearing in (9) and (10) of the combinatorial properties of a matrix A of m rows and n columns, all of whose entries are 0's and l's. Let the sum of row i of A be denoted by r i and let the sum of column j of A be denoted by S j . We call R = (r 1 , … , r m ) the row sum vector and S = (s 1 . . , s n ) the column sum vector of A . The vectors R and S determine a class 1.1 consisting of all (0, 1)-matrices of m rows and n columns, with row sum vector R and column sum vector S. The majorization concept yields simple necessary and sufficient conditions on R and S in order that the class 21 be non-empty (4; 9). Generalizations of this result and a critical survey of a wide variety of related problems are available in (6).

This paper continues a study appearing in (5) of the combinatorial properties of 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 and let the sum of column i of A be noted by s t . We call R = ( r 1 , … , r m ) the row sum vector and S = ( s 1 , … , s n ) the column sum vector of A . The vectors R and S determine a class consisting of all (0, 1)-matrices of m rows and n columns, with row sum vector R and column sum vector S. Simple arithmetic properties of R and S are necessary and sufficient for the existence of a class (1 ; 5).

1. Introduction. Let Q be a matrix of order v, all of whose entries are 0's and l's. Let the total number of l's in Q be t , and let the absolute value of the determinant of Q be denoted by |det Q |. In this paper we study the problem of determining the maximum of |det Q | for fixed t and v. It turns out that this problem is closely related to the v , k , λ problem, which has been extensively studied of late.