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A180512 Triangle of the number of alternating sign matrices according to the number of -1's.

%I A180512 #36 Jul 22 2025 08:27:52
%S A180512 1,2,6,1,24,16,2,120,200,94,14,1,720,2400,2684,1284,310,36,2,5040,
%T A180512 29400,63308,66158,38390,13037,2660,328,26,1
%N A180512 Triangle of the number of alternating sign matrices according to the number of -1's.
%C A180512 The first column is the factorial, A000142.
%C A180512 The second column forms coefficients of Laguerre polynomials, A001810.
%C A180512 From _Arvind Ayyer_, Mar 15 2018: (Start)
%C A180512 Consider the row generating function A_n(x) = sum_k a(n,k) x^k. Then
%C A180512 A_n(0) = n!, A000142.
%C A180512 A_n(1) = number of ASM's, A005130.
%C A180512 A_n(2) = number of domino tilings of the Aztec diamond, A006125.
%C A180512 A_n(3) = 3-enumeration of n X n alternating-sign matrices, A059477. (End)
%H A180512 FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/St000065/">The number of entries equal to negative one in the alternating sign matrix</a>
%H A180512 Florent Le Gac, <a href="http://www.theses.fr/2011BOR14287">Quelques problèmes d’énumération autour des matrices à signes alternants</a>, thesis, LaBRI Bordeaux, 2011.
%H A180512 Wikipedia, <a href="http://en.wikipedia.org/wiki/Alternating_sign_matrix">Alternating Sign Matrix</a>
%e A180512 In triangular format, the numbers of ASMs is as follows:
%e A180512 n=1:1
%e A180512 n=2:2
%e A180512 n=3:6,1
%e A180512 n=4:24,16,2
%e A180512 n=5:120,200,94,14,1
%e A180512 n=6:720,2400,2684,1284,310,36,2
%e A180512 n=7:5040,29400,63308,66158,38390,13037,2660,328,26,1
%Y A180512 Row sums are A005130
%Y A180512 Cf. A000142, A006125, A059477, A001810.
%K A180512 nonn,hard,tabf
%O A180512 1,2
%A A180512 _Arvind Ayyer_, Jan 20 2011
%E A180512 T(7, 7) corrected by _Arvind Ayyer_, Feb 12 2018