科研之友
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摘要
Let s, t, m, n be positive integers such that sm = tn. Let M(m,s; n, t) be the number of m x n matrices over {0, 1, 2,...} with each row summing to s and each column summing to t. Equivalently, M(m,s;n,t) counts 2-way contingency tables of order m x n such that the row marginal sums are all s and the column marginal sums are all t. A third equivalent description is that M (in, s; n,t) is the number of semiregular labelled bipartite multigraphs with m vertices of degree s and n vertices of degree t. When m=n and s=t such matrices are also referred to as nxn magic or semimagic squares with line sums equal to t. We prove a precise asymptotic formula for M(m,s;n, t) which is valid over a range of (m,s;n,t) in which m, n -> infinity while remaining approximately equal and the average entry is not too small. This range includes the case where m/n, n/m, s/n and t/m are bounded from below.
- 出版日期2010
