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

Abstract

We wish to consider the class of (0,1)-matrices with prescribed row and column sums. LetR= (r1,r2, …,rm) andS= (s1,s2, …,sn) be vectors with nonnegative integral entries andr1+r2+ … +rm=s1+s2+ … +sn. We define the classto be the set ofm×n(0, 1)-matrices withithrow sumriandjthcolumn sumsjfor 1 ≦i≦mand 1 ≦j≦n.Gale and Ryser independently found simple necessary and sufficient conditions forto be nonempty [9,14]. FromR, we form anm×n(0, 1)-matrixĀas follows. Theith row sum ofĀisriand the 1‘s are as far to the left as possible. Letbe thejthcolumn sum ofA. We define the sequenceto be theconjugateof the sequence (ri).