On the enumeration of rectangular (0, 1)-matrices

Abstract

The construction and enumeration of (0, 1)-matrices with given line-sums is described for the rectangular cases often encountered in applications. Improved approximations are provided for the number of such matrices. Some new enumeration results for semi-regular bipartite graphs are included, and the related category of the quasi-semiregular bipartite graphs is recognized. The range of certain elements of products of a (0, I)-matrix is considered as a function of the line-sums. This, in turn, is related to the range in the numbers of interchanges available. Improvements in statistical practice that come from these constructions and enumerations are described.

Keywords:

• (0,1)-matrices
• enumeration
• Approximations
• bipartite graph
• semi-regular bipartite graphs
• random graphs
• Rasch model
• cluster analysis
• similarity
• statistical ecology