Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Asymptotic Enumeration of Dense 0-1 Matrices with Equal Row Sums and Equal Column Sums. | Asymptotic Enumeration of Dense 0-1 Matrices with Equal Row Sums and Equal Column Sums E. Rodney Canfield Department of Computer Science University of Georgia Athens GA 30602 UsA erc@ Brendan D. McKay 1 Department of Computer Science Australian National University Canberra ACT 0200 Australia bdm@ Submitted Dec 22 2004 Accepted Jun 11 2005 Published Jun 19 2005 Mathematics Subject Classifications 05A16 05C30 62H17 Abstract Let s t m n be positive integers such that sm tn. Let B m s n t be the number of m X n matrices over 0 1 with each row summing to s and each column summing to t. Equivalently B m s n t is the number of semiregular bipartite graphs with m vertices of degree s and n vertices of degree t. Define the density A s n t m. The asymptotic value of B m s n t has been much studied but the results are incomplete. McKay and Wang 2003 solved the sparse case A 1 A o mn 1 using combinatorial methods. In this paper we use analytic methods to solve the problem for two additional ranges. In one range the matrix is relatively square and the density is not too close to 0 or 1. In the other range the matrix is far from square and the density is arbitrary. Interestingly the asymptotic value of B m s n t can be expressed by the same formula in all cases where it is known. Based on computation of the exact values for all m n 30 we conjecture that the same formula holds whenever m n 1 regardless of the density. 1 Introduction Let s t m n be positive integers such that sm tn. Let B m s n t be the number of m X n matrices over 0 1 with each row summing to s and each column summing to t. Equivalently B m s n t is the number of semiregular bipartite graphs with m vertices of degree s and n vertices of degree t. The density A s n t m is the fraction of entries in the matrix which are 1. Research supported by the NSA Mathematical Sciences Program 1 Research supported by the Australian Research Council THE ELECTRONIC JOURNAL OF COMBINATORICS 12 2005 R29 1 .
