Matrices of zeros and ones with fixed row and column sum vectors

Abstract

Let m and n be positive integers, and let R=( r 1,…, r m ) and S=( s 1,…, s n ) be nonnegative integral vectors. We survey the combinational properties of the set of all m × n matrices of 0's and 1's having r i 1's in row i and s i 1's in column j. A number of new results are proved. The results can be also be formulated in terms of a set of bipartite graps with bipartition into m and n vertices having degree sequence R and S, respectively. They can also be formulated in terms of the set of hypergraphs with m vertices having degree sequence R and n edges whose cardinalities are given by S.