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%I #54 Jun 13 2021 07:12:35
%S 1,1,1,2,3,2,7,14,14,7,42,105,135,105,42,429,1287,2002,2002,1287,429,
%T 7436,26026,47320,56784,47320,26026,7436,218348,873392,1813968,
%U 2519400,2519400,1813968,873392,218348,10850216,48825972,113927268,179028564
%N 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)
%C 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.
%C Named after the American mathematician David Peter Robbins (1942-2003). - _Amiram Eldar_, Jun 13 2021
%D David Bressoud, Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture, Cambridge University Press, 1999, p. 5.
%H N. J. A. Sloane, <a href="/A048601/b048601.txt">Table of n, a(n) for n = 1..1275</a> [Rows 1..50, flattened]
%H Roger E. Behrend, Philippe Di Francesco and Paul Zinn-Justin, <a href="http://arxiv.org/abs/1103.1176">On the weighted enumeration of Alternating Sign Matrices and Descending Plane Partitions</a>, arXiv:1103.1176 [math.CO], 2011.
%H David Bressoud and James Propp, <a href="http://www.ams.org/notices/199906/fea-bressoud.pdf">How the alternating sign matrix conjecture was solved</a>, Notices Amer. Math. Soc., Vol. 46, No. 6 (1999), p. 637-646.
%H FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/StatisticsDatabase/St000066">The column of the unique '1' in the first row of the alternating sign matrix</a>.
%H FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/St000066/">The column of the unique 1 in the first row of the alternating sign matrix</a>.
%H P. Di Francesco, <a href="http://arXiv.org/abs/cond-mat/0409576">A refined Razumov-Stroganov conjecture II</a>, arXiv:cond-mat/0409576 [cond-mat.stat-mech], 2004.
%H D. Gerdemann, <a href="https://www.youtube.com/watch?v=fxJpxvziJYs">Robbins Triangle for Counting Alternating Sign Matrices</a> YouTube Video, 2015.
%H W. H. Mills, David P. Robbins and Howard Rumsey, Jr., <a href="http://dx.doi.org/10.1016/0097-3165(83)90068-7">Alternating sign matrices and descending plane partitions</a> J. Combin. Theory, Ser. A, Vol. 34, No. 3 (1983), pp. 340-359. MR0700040 (85b:05013).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AlternatingSignMatrix.html">Alternating Sign Matrix</a>.
%H Doron Zeilberger, <a href="http://arXiv.org/abs/math.CO/9606224">Proof of the Refined Alternating Sign Matrix Conjecture</a>, arXiv:math/9606224 [math.CO], 1996.
%H Doron Zeilberger, <a href="http://www.math.rutgers.edu/~zeilberg/DaveRobbins/guess.html">Dave Robbins's Art of Guessing</a>, Adv. in Appl. Math., Vol. 34 (2005), pp. 939-954.
%F 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);
%e Triangle begins:
%e 1,
%e 1, 1,
%e 2, 3, 2,
%e 7, 14, 14, 7,
%e 42, 105, 135, 105, 42,
%e 429, 1287, 2002, 2002, 1287, 429,
%e 7436, 26026, 47320, 56784, 47320, 26026, 7436,
%e ...
%p 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);
%t 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 *)
%Y Row sums (also borders) of triangle give A005130. Cf. A051106.
%Y A210697 is a companion triangle.
%K nonn,tabl,nice,easy,look
%O 1,4
%A _N. J. A. Sloane_
%E More terms from _James A. Sellers_
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