Asymptotic enumeration of integer matrices with large equal row and column sums

• Published in: Combinatorica (Budapest. 1981) Vol. 30; no. 6; pp. 655 - 680
• Main Authors: Rodney Canfield, E., McKay, Brendan D.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01-11-2010
• Subjects: Combinatorics; Mathematics; Mathematics and Statistics; 05A16; 62H17; 05C30
• ISSN: 0209-9683; 1439-6912
• DOI: 10.1007/s00493-010-2426-1

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.