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

1, 2, 6, 1, 24, 16, 2, 120, 200, 94, 14, 1, 720, 2400, 2684, 1284, 310, 36, 2, 5040, 29400, 63308, 66158, 38390, 13037, 2660, 328, 26, 1

Offset: 1

Author

Arvind Ayyer, Jan 20 2011

Keywords

• nonn
• hard
• tabf

Comments

The first column is the factorial, A000142.

The second column forms coefficients of Laguerre polynomials, A001810.

From Arvind Ayyer, Mar 15 2018: (Start)

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

A_n(0) = n!, A000142.

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

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

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

Examples

			In triangular format, the numbers of ASMs is as follows:
n=1:1
n=2:2
n=3:6,1
n=4:24,16,2
n=5:120,200,94,14,1
n=6:720,2400,2684,1284,310,36,2
n=7:5040,29400,63308,66158,38390,13037,2660,328,26,1
		

Links

• FindStat - Combinatorial Statistic Finder, The number of entries equal to negative one in the alternating sign matrix
• Florent Le Gac, Quelques problèmes d’énumération autour des matrices à signes alternants, thesis, LaBRI Bordeaux, 2011.
• Wikipedia, Alternating Sign Matrix

Crossrefs

Row sums are A005130

Cf. A000142, A006125, A059477, A001810.

Extensions

T(7, 7) corrected by Arvind Ayyer, Feb 12 2018