|
|
A260680
|
|
Peaceable coexisting armies of queens: number of inequivalent configurations with maximum number of queens as given in A250000.
|
|
2
|
|
|
|
OFFSET
|
1,4
|
|
COMMENTS
|
Two solutions are regarded as equivalent if one can be obtained from the other by rotations, reflections, interchanging the colors (a group of order 16).
I used two computational methods, both implemented via PROC OPTMODEL from SAS:
One round of constraint programming, with LEXICO constraints to account for symmetry and an option to generate all solutions. This method returns only the lexicographically smallest representative of each equivalence class.
Multiple rounds of integer linear programming, with 16 additional cuts (one per group element) after each solution is found, to avoid generating an equivalent solution. This method terminates when the resulting cuts make the problem infeasible.
The attached text files are from the second method. (End)
|
|
LINKS
|
|
|
EXAMPLE
|
For n = 3, a(3) = 1 because the following solution is unique up to equivalence:
-----
|W..|
|...|
|.B.|
-----
n=4:
----------------------------------------------------------
|..B.||.B..||.B..||....||.BB.||..B.||...W||..B.|..B.|..W.|
|....||.B..||...B||.B.B||....||.B..||.B..||...B|B...|B...|
|...B||....||....||....||....||...W||..B.||.W..|...W|...B|
|WW..||W.W.||W.W.||W.W.||W..W||W...||W...||W...|.W..|.W..|
----------------------------------------------------------
n=5:
---------------------
|W...W||..B.B||.W.W.|
|..B..||W....||..W..|
|.B.B.||..B.B||B...B|
|..B..||W....||..W..|
|W...W||.W.W.||B...B|
---------------------
(End)
From Rob Pratt, Mar 18 2019, additional solution for n=6 (not covered in attached pdf):
--------
|....W.|
|...W.W|
|B.....|
|B.B...|
|....WW|
|B.B...|
--------
|
|
CROSSREFS
|
|
|
KEYWORD
|
hard,nonn,more
|
|
AUTHOR
|
|
|
EXTENSIONS
|
a(4), a(6), and a(8) corrected by Rob Pratt, Mar 18 2019
|
|
STATUS
|
approved
|
|
|
|