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A143670 Array of higher spin alternating sign matrices, read by antidiagonals. 1
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,5
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.
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
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)!.
EXAMPLE
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..|
========================================================
CROSSREFS
Sequence in context: A346083 A301922 A144510 * A169730 A220725 A333142
KEYWORD
more,nonn,tabl
AUTHOR
Jonathan Vos Post, Aug 28 2008
EXTENSIONS
Some terms of the 7th diagonal from R. J. Mathar, Mar 04 2010
STATUS
approved

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Last modified April 18 09:20 EDT 2024. Contains 371769 sequences. (Running on oeis4.)