OFFSET
1,4
COMMENTS
See Mills et al., pp. 353-354 and 359 for precise definition. As of 1983 no formula was known for these numbers.
These are the values of a bivariate generating function for the ASMs by numbers of entries equal to -1 and by position of 1 in the first row (see Example section). Here weight x=3 is chosen, giving a decomposition of the 3-enumeration of the n X n ASMs.
As a triangle of coefficients of polynomials, A210697 has interesting properties relating the (2n+1)-th row and the n-th row (see Mills et al., p. 359).
LINKS
W. H. Mills, David P Robbins, Howard Rumsey Jr., Alternating sign matrices and descending plane partitions J. Combin. Theory Ser. A 34 (1983), no. 3, 340--359. MR0700040 (85b:05013). See p. 359.
EXAMPLE
The bivariate g.f. as a table of polynomials.
(degree of x is the count of -1 entries in the ASM)
Setting x=k gives the k-enumeration of the ASMs
n
1 | 1
2 | 1, 1
3 | 2, 2+x, 2
4 | 6+x, 6+7*x+x^2, 6+7*x+x^2, 6+x
5 | 24 + 16*x + 2*x^2, 24 + 52*x + 26*x^2 + 3*x^3, 24 + 64*x + 38*x^2 +
| 8*x^3 + x^4, 24 + 52*x + 26*x^2 + 3*x^3, 24 + 16*x + 2*x^2
...
Triangle begins:
n
1 | 1
2 | 1 1
3 | 2 5 2
4 | 9 36 36 9
5 | 90 495 855 495 90
6 | 2025 14175 34830 34830 14175 2025
...
CROSSREFS
KEYWORD
AUTHOR
N. J. A. Sloane, Mar 30 2012
EXTENSIONS
More terms, definitions and examples by Olivier Gérard, Apr 02 2015
STATUS
approved