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A048601 Robbins triangle read by rows: T(n,k) = number of alternating sign n X n matrices with a 1 at top of column k (n >= 1, 1<=k<=n) 10
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, 218348, 873392, 1813968, 2519400, 2519400, 1813968, 873392, 218348, 10850216, 48825972, 113927268, 179028564 (list; table; graph; refs; listen; history; text; internal format)
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.
P. Di Francesco, A refined Razumov-Stroganov conjecture II, arXiv:cond-mat/0409576 [cond-mat.stat-mech], 2004.
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
Row sums (also borders) of triangle give A005130. Cf. A051106.
A210697 is a companion triangle.
Sequence in context: A271322 A170842 A014784 * A008317 A139011 A338770
KEYWORD
nonn,tabl,nice,easy,look
AUTHOR
EXTENSIONS
More terms from James A. Sellers
STATUS
approved

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Last modified March 19 07:25 EDT 2024. Contains 370955 sequences. (Running on oeis4.)