A180512 - OEIS

%I #34 Mar 15 2018 03:01:31

%S 1,2,6,1,24,16,2,120,200,94,14,1,720,2400,2684,1284,310,36,2,5040,

%T 29400,63308,66158,38390,13037,2660,328,26,1

%N Triangle of the number of alternating sign matrices according to the number of -1's

%C The first column is the factorial, A000142.

%C The second column forms coefficients of Laguerre polynomials, A001810.

%C From _Arvind Ayyer_, Mar 15 2018: (Start)

%C Consider the row generating function A_n(x) = sum_k a(n,k) x^k. Then

%C A_n(0) = n!, A000142.

%C A_n(1) = number of ASM's, A005130.

%C A_n(2) = number of domino tilings of the Aztec diamond, A006125.

%C A_n(3) = 3-enumeration of n X n alternating-sign matrices, A059477. (End)

%H 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 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 Wikipedia, <a href="http://en.wikipedia.org/wiki/Alternating_sign_matrix">Alternating Sign Matrix</a>

%e In triangular format, the numbers of ASMs is as follows:

%e n=1:1

%e n=2:2

%e n=3:6,1

%e n=4:24,16,2

%e n=5:120,200,94,14,1

%e n=6:720,2400,2684,1284,310,36,2

%e n=7:5040,29400,63308,66158,38390,13037,2660,328,26,1

%Y Row sums are A005130

%Y Cf. A000142, A006125, A059477, A001810.

%K nonn,hard,tabf

%O 1,2

%A _Arvind Ayyer_, Jan 20 2011

%E T(7, 7) corrected by _Arvind Ayyer_, Feb 12 2018