A260680 Peaceable coexisting armies of queens: number of inequivalent configurations with maximum number of queens as given in A250000.

1, 1, 1, 10, 3, 35, 19, 71, 18, 380

Offset: 1

Author

Christian Schroeder, Nov 15 2015

Keywords

• hard
• nonn
• more

Comments

From Rob Pratt, Apr 05 2019: (Start)

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)

Examples

			For n = 3, a(3) = 1 because the following solution is unique up to equivalence:
-----
|W..|
|...|
|.B.|
-----
From _Rob Pratt_ in A250000, Nov 30 2014 thru Jul 29 2015: (Start)
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...|
--------
		

Links

• Luca Petrone, Graphic illustrations of a(6) and a(7)
• Rob Pratt, Solutions for n = 3
• Rob Pratt, Solutions for n = 4
• Rob Pratt, Solutions for n = 5
• Rob Pratt, Solutions for n = 6
• Rob Pratt, Solutions for n = 7
• Rob Pratt, Solutions for n = 8
• Rob Pratt, Solutions for n = 9
• Rob Pratt, Solutions for n = 10

Crossrefs

Cf. A250000.

Extensions

a(6)-a(8) from Luca Petrone, Mar 11 2016

a(4), a(6), and a(8) corrected by Rob Pratt, Mar 18 2019

a(9) and a(10) from Rob Pratt, Mar 19 2019