OFFSET
1,4
COMMENTS
An alternating sign matrix is a matrix of 0's and 1's such that (a) the sum of each row and column is 1; (b) the nonzero entries in each row and column alternate in sign.
Named after the American mathematician David Peter Robbins (1942-2003). - Amiram Eldar, Jun 13 2021
REFERENCES
David Bressoud, Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture, Cambridge University Press, 1999, p. 5.
LINKS
N. J. A. Sloane, Table of n, a(n) for n = 1..1275 [Rows 1..50, flattened]
Roger E. Behrend, Philippe Di Francesco and Paul Zinn-Justin, On the weighted enumeration of Alternating Sign Matrices and Descending Plane Partitions, arXiv:1103.1176 [math.CO], 2011.
David Bressoud and James Propp, How the alternating sign matrix conjecture was solved, Notices Amer. Math. Soc., Vol. 46, No. 6 (1999), p. 637-646.
FindStat - Combinatorial Statistic Finder, The column of the unique '1' in the first row of the alternating sign matrix.
FindStat - Combinatorial Statistic Finder, The column of the unique 1 in the first row of the alternating sign matrix.
P. Di Francesco, A refined Razumov-Stroganov conjecture II, arXiv:cond-mat/0409576 [cond-mat.stat-mech], 2004.
D. Gerdemann, Robbins Triangle for Counting Alternating Sign Matrices YouTube Video, 2015.
W. H. Mills, David P. Robbins and Howard Rumsey, Jr., Alternating sign matrices and descending plane partitions J. Combin. Theory, Ser. A, Vol. 34, No. 3 (1983), pp. 340-359. MR0700040 (85b:05013).
Eric Weisstein's World of Mathematics, Alternating Sign Matrix.
Doron Zeilberger, Proof of the Refined Alternating Sign Matrix Conjecture, arXiv:math/9606224 [math.CO], 1996.
Doron Zeilberger, Dave Robbins's Art of Guessing, Adv. in Appl. Math., Vol. 34 (2005), pp. 939-954.
FORMULA
T(n,k) = binomial(n+k-2, k-1)*((2*n-k-1)!/(n-k)!) * product(((3*j+1)!/(n+j)!), j=0..n-2);
EXAMPLE
Triangle begins:
1,
1, 1,
2, 3, 2,
7, 14, 14, 7,
42, 105, 135, 105, 42,
429, 1287, 2002, 2002, 1287, 429,
7436, 26026, 47320, 56784, 47320, 26026, 7436,
...
MAPLE
T:=(n, k)-> binomial(n+k-2, k-1)*((2*n-k-1)!/(n-k)!)*product(((3*j+1)!/(n+j)!), j=0..n-2);
MATHEMATICA
t[n_, k_] := Binomial[n+k-2, k-1]*((2*n-k-1)!/(n-k)!)*Product[((3*j+1)!/(n+j)!), {j, 0, n-2}]; Table[t[n, k], {n, 1, 9}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 12 2012, from formula *)
CROSSREFS
AUTHOR
EXTENSIONS
More terms from James A. Sellers
STATUS
approved