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Properties of a Class of (0,1)-Matrices Covering a given Matrix

Published online by Cambridge University Press:  20 November 2018

R. P. Anstee
R. P. Anstee*
Affiliation:
University of Waterloo, Waterloo, Ontario
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We wish to consider the class of (0,1)-matrices with prescribed row and column sums. Let R = (r 1, r 2, …, rm ) and S= (s 1, s 2, …, sn ) be vectors with nonnegative integral entries and r 1 + r 2 + … + rm = s 1 + s 2 + … + sn . We define the class to be the set of m × n (0, 1)-matrices with i th row sum ri and j th column sum sj for 1 ≦ i ≦ m and 1 ≦ j ≦ n.

Gale and Ryser independently found simple necessary and sufficient conditions for to be nonempty [9 , 14]. From R, we form an m × n (0, 1)-matrix Ā as follows. The ith row sum of Ā is ri and the 1‘s are as far to the left as possible. Let be the j th column sum of A. We define the sequence to be the conjugate of the sequence (ri).

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 34 , Issue 2 , 01 April 1982 , pp. 438 - 453
Copyright
Copyright © Canadian Mathematical Society 1982

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