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A260319
Number of singly symmetric characteristic solutions to the n-queens problem.
4
0, 0, 0, 0, 0, 1, 2, 1, 4, 3, 12, 18, 32, 105, 310, 734, 2006, 4526, 11046, 36035, 93740, 312673, 895310, 2917938, 8532332, 28929567
OFFSET
1,7
COMMENTS
The problem of placing eight queens on a chessboard so that no one of them can take any other in a single move is a particular case of the more general problem: On a square array of n X n cells place n objects, one on each of n different cells, in such a way that no two of them lie on the same row, column, or diagonal.
There are no centrosymmetric solutions for n < 6, if by "centrosymmetric" we exclude "doubly symmetric" cases; and there is just one complete set for n = 6: 246135, 362514, 415263, 531642.
On the ordinary chessboard of 8 X 8 cells there are a total of 92 solutions, consisting of 11 sets of equivalent ordinary solutions and one set of equivalent symmetric solutions. There are no doubly symmetric solutions in this case. These sets may be generated in the ordinary case by 15863724, 16837425, 24683175, 2571384, 25741863, 26174835, 26831475, 27368514, 27581463, 35841726, 36258174 and in the symmetric case by 35281746.
REFERENCES
Maurice Kraitchik: Mathematical Recreations. Mineola, NY: Dover, 2nd ed. 1953, p. 247-255 (The Problem of the Queens).
LINKS
P. Capstick and K. McCann, The problem of the n queens, apparently unpublished, no date (circa 1990?) [Scanned copy]
M. A. Sainte-Laguë, Les Réseaux (ou Graphes), Mémorial des Sciences Mathématiques, Fasc. 18, Gauthier-Villars, Paris, 1923, 64 pages. See p. 47.
M. A. Sainte-Laguë, Les Réseaux (ou Graphes), Mémorial des Sciences Mathématiques, Fasc. 18, Gauthier-Villars, Paris, 1923, 64 pages. See p. 47. [Incomplete annotated scan of title page and pages 18-51]
FORMULA
a(n) = -A000170(n)/4 + 2*A002562(n) - 3*A260318(n)/2, n > 1. - R. J. Mathar, Jul 24 2015
CROSSREFS
KEYWORD
nonn,more
AUTHOR
N. J. A. Sloane, Jul 22 2015
EXTENSIONS
Name edited (by inserting "singly", since "doubly symmetric" solutions are symmetric but not counted here) by Don Knuth, Mar 25 2022
STATUS
approved