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).

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• tabl
• nice
• easy
• look

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

Examples

			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,
  ...
		

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.

Crossrefs

Row sums (also borders) of triangle give A005130. Cf. A051106.

A210697 is a companion triangle.

Programs

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 *)

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);

Extensions

More terms from James Sellers