An alternating sign matrix is a matrix of 0s, 1s, and s
in which the entries in each row or column sum to 1 and the nonzero entries in each
row and column alternate in sign. The first few for , 2, ... are shown below:
(1)
(2)
(3)
(4)
Such matrices satisfy the additional property that s in a row or column must have a "outside" it (i.e., all s are "bordered" by s). The numbers of alternating sign matrices for , 2, ... are given by 1, 2, 7, 42, 429, 7436, 218348, ...
(OEIS A005130).
The conjecture that the number of is explicitly given by the formula
Making a triangular array of the number of with a 1 at the top of column gives
(9)
(OEIS A048601), and taking the ratios of adjacent
terms gives the array
(10)
(OEIS A029656 and A029638). The fact that these numerators and denominators are respectively the numbers in the
(2, 1)- and (1, 2)-Pascal triangles which are different from 1 is known as the refined alternating sign matrix
conjecture.
Andrews, G. E. "Plane Partitions (III): The Weak Macdonald Conjecture." Invent. Math.53, 193-225, 1979.Bressoud,
D. Proofs
and Confirmations: The Story of the Alternating Sign Matrix Conjecture. Cambridge,
England: Cambridge University Press, 1999.Bressoud, D. and Propp, J.
"How the Alternating Sign Matrix Conjecture was Solved." Not. Amer.
Math. Soc.46, 637-646.Finch, S. R. Mathematical
Constants. Cambridge, England: Cambridge University Press, p. 413, 2003.Kuperberg,
G. "Another Proof of the Alternating-Sign Matrix Conjecture." Internat.
Math. Res. Notes, No. 3, 139-150, 1996.Mills, W. H.; Robbins,
D. P.; and Rumsey, H. Jr. "Proof of the Macdonald Conjecture." Invent.
Math.66, 73-87, 1982.Mills, W. H.; Robbins, D. P.;
and Rumsey, H. Jr. "Alternating Sign Matrices and Descending Plane Partitions."
J. Combin. Th. Ser. A34, 340-359, 1983.Pickover, C. A.
"Princeton Numbers." Ch. 79 in Wonders
of Numbers: Adventures in Mathematics, Mind, and Meaning. Oxford, England:
Oxford University Press, pp. 189-192, 2001.Robbins, D. P.
"The Story of 1, 2, 7, 42, 429, 7436, ...." Math. Intell.13,
12-19, 1991.Robbins, D. P. and Rumsey, H. Jr. "Determinants
and Alternating Sign Matrices." Adv. Math.62, 169-184, 1986.Sloane,
N. J. A. Sequences A005130/M1808,
A029638, A029656,
A048601, and A050204
in "The On-Line Encyclopedia of Integer Sequences."Stanley,
R. P. "A Baker's Dozen of Conjectures Concerning Plane Partitions."
In Combinatoire
Énumérative. Proceedings of the colloquium held at the Université
du Québec, Montreal, May 28-June 1, 1985 (Ed. G. Labelle and
P. Leroux). New York: Springer-Verlag, pp. 285-293, 1986.Zeilberger,
D. "Proof of the Alternating Sign Matrix Conjecture." Electronic J.
Combinatorics3, No. 2, R13, 1-84, 1996. http://www.combinatorics.org/Volume_3/Abstracts/v3i2r13.html.Zeilberger,
D. "Proof of the Refined Alternating Sign Matrix Conjecture." New York
J. Math.2, 59-68, 1996.Zeilberger, D. "A Constant Term
Identity Featuring the Ubiquitous (and Mysterious) Andrews-Mills-Robbins-Rumsey numbers
1, 2, 7, 42, 429, ...." J. Combin. Theory A66, 17-27, 1994.
s
in which the entries in each row or column sum to 1 and the nonzero entries in each
row and column alternate in sign. The first few for 
, 2, ... are shown below:
![[1]](/images/equations/AlternatingSignMatrix/Inline5.svg)
![[1 0; 0 1],[0 1; 1 0]](/images/equations/AlternatingSignMatrix/Inline8.svg)
![[ 0 0 1; 0 1 0; 1 0 0],[ 0 0 1; 1 0 0; 0 1 0],[ 0 1 0; 0 0 1; 1 0 0],[ 0 1 0; 1 -1 1; 0 1 0],](/images/equations/AlternatingSignMatrix/Inline11.svg)
![[ 0 1 0; 1 0 0; 0 0 1],[ 1 0 0; 0 0 1; 0 1 0],[ 1 0 0; 0 1 0; 0 0 1]](/images/equations/AlternatingSignMatrix/Inline14.svg)
s in a row or column must have a 
"outside" it (i.e., all 
s are "bordered" by 
s). The numbers of 
alternating sign matrices 
for 
, 2, ... are given by 1, 2, 7, 42, 429, 7436, 218348, ...
(OEIS A005130).
of 
is explicitly given by the formula
can be expressed in closed form as a complicated function
of Barnes G-functions, but additional simplification
is likely possible.
is given by
is the gamma function.
be the number of 
alternating sign matrices with one in the top row occurring
in the 
th
position. Then
implies (7) (Mills et al. 1983).
with a 1 at the top of column 
gives
