TY - JOUR
T1 - Asymptotic enumeration of integer matrices with large equal row and column sums
AU - Canfield, E. Rodney
AU - McKay, Brendan D.
PY - 2010/11
Y1 - 2010/11
N2 - Let s, t, m, n be positive integers such that sm = tn. Let M(m, s; n, t) be the number of m×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×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(m, 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 n×n 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→∞ 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.
AB - Let s, t, m, n be positive integers such that sm = tn. Let M(m, s; n, t) be the number of m×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×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(m, 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 n×n 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→∞ 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.
UR - http://www.scopus.com/inward/record.url?scp=79952793386&partnerID=8YFLogxK
U2 - 10.1007/s00493-010-2426-1
DO - 10.1007/s00493-010-2426-1
M3 - Article
SN - 0209-9683
VL - 30
SP - 655
EP - 680
JO - Combinatorica
JF - Combinatorica
IS - 6
ER -