A143670 Array of higher spin alternating sign matrices, read by antidiagonals.

1, 1, 1, 1, 2, 1, 1, 7, 3, 1, 1, 42, 26, 4, 1, 1, 429, 628, 70, 5, 1, 1, 7436, 41784, 5102, 155, 6, 1, 1, 218348, 7517457, 1507128, 28005

Offset: 1

Author

Jonathan Vos Post, Aug 28 2008

Keywords

• more
• nonn
• tabl

Comments

Adapted from Table 1, p. 5: |ASM(n, r)|, where A[k,n] = |ASM(n, k)|. Abstract: We define a higher spin alternating sign matrix to be an integer-entry square matrix in which, for a nonnegative integer r, all complete row and column sums are r and all partial row and column sums extending from each end of the row or column are nonnegative. Such matrices correspond to configurations of spin r/2 statistical mechanical vertex models with domain-wall boundary conditions.

The case r = 1 gives standard alternating sign matrices, while the case in which all matrix entries are nonnegative gives semimagic squares. We show that the higher spin alternating sign matrices of size n are the integer points of the r-th dilate of an integral convex polytope of dimension (n-1)^2 whose vertices are the standard alternating sign matrices of size n. It then follows that, for fixed n, these matrices are enumerated by an Ehrhart polynomial in r.

Examples

			The array begins:
========================================================
....|.r=0|..r=1.|.....r=2.|.......r=3.|..........r=4.|
n=1.|..1.|...1..|......1..|.........1.|...........1..|.A000012
n=2.|..1.|...2..|......3..|.........4.|...........5..|.A000027
n=3.|..1.|...7..|.....26..|........70.|.........155..|
n=4.|..1.|..42..|....628..|......5102.|.......28005..|
n=5.|..1.|.429..|..41784..|...1507128.|....28226084..|
n=6.|..1.|7436..|7517457..|1749710096.|152363972022..|
========================================================
		

Links

• Roger E. Behrend and Vincent A. Knight, Higher Spin Alternating Sign Matrices, arXiv:0708.2522 [math.CO], 2007.
• Roger E. Behrend, Vincent A. Knight, Higher Spin Alternating sign matrices, El. J. Combinat 14 (2007) #R83

Crossrefs

Cf. A000012, A000027, A005130.

Formula

Apart from the trivial formulas |ASM(0, n)| = 1 (since ASM(0, n) contains only the n X n zero matrix), |ASM(1, r)| = 1 and |ASM(2, r)| = r+1, the only previously- known formula for a special case of |ASM(n, r)| is |ASM(n, 1)| = Sum_{i=0..n-1} (3*i+1)!/(n+1)!.

Extensions

Some terms of the 7th diagonal from R. J. Mathar, Mar 04 2010