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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.

REFERENCES

D. 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]

R. E. Behrend, P. Di Francesco, P. Zinn-Justin, On the weighted enumeration of Alternating Sign Matrices and Descending Plane Partitions, arXiv:1103.1176  [math.CO], 2011.

D. M. Bressoud and J. Propp, How the alternating sign matrix conjecture was solved, Notices Amer. Math. Soc., 46 (No. 6, 1999), 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, Howard Rumsey Jr., Alternating sign matrices and descending plane partitions J. Combin. Theory Ser. A 34 (1983), no. 3, 340--359. MR0700040 (85b:05013).

Eric Weisstein's World of Mathematics, Alternating Sign Matrix.

D. Zeilberger, Proof of the Refined Alternating Sign Matrix Conjecture, arXiv:math/9606224 [math.CO], 1996.

D. Zeilberger, Dave Robbins's Art of Guessing, Adv. in Appl. Math. 34 (2005), 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-Fran├žois 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 A063708

Adjacent sequences:  A048598 A048599 A048600 * A048602 A048603 A048604

KEYWORD

nonn,tabl,nice,easy,look

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from James A. Sellers

STATUS

approved

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Last modified August 19 01:29 EDT 2017. Contains 290787 sequences.