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A071706
Number of complete mappings f(x) of the cyclic group Z_{2n+1} such that -f(-x)=f.
2
1, 1, 3, 5, 21, 69, 319, 1957, 12513, 85445, 656771, 5591277, 51531405, 509874417, 5438826975, 62000480093, 752464463029, 9685138399785, 131777883431119
OFFSET
0,3
COMMENTS
A complete mapping of a cyclic group (Zn,+) is a permutation f(x) of Zn such that f(0)=0 and that f(x)-x is also a permutation.
a(n) is the number of complete mappings fixed under rotation R180 where R180(f)(x)=-f(-x). This sequence (n) equals TSQ_R180(n), the number of solutions of the toroidal n-queen problem fixed under rotation R180. A solution of toroidal-semi n-queen problem is a permutation f(x) of Zn such that f(x)-x is also a permutation.
REFERENCES
Y. P. Shieh, "Partition strategies for #P-complete problems with applications to enumerative combinatorics", PhD thesis, National Taiwan University, 2001.
Y. P. Shieh, J. Hsiang and D. F. Hsu, "On the enumeration of Abelian k-complete mappings", vol. 144 of Congressus Numerantium, 2000, pp. 67-88.
EXAMPLE
f(x)=6x in (Z7,+) is a complete mapping of Z7 since f(0)=0 and f(x)-x (=5x) is also a permutation of Z7. R180(f)(x)=-f(-x) (=6x). So f(x) is fixed under R180.
CROSSREFS
Sequence in context: A264683 A286033 A056803 * A196418 A189560 A216385
KEYWORD
nonn
AUTHOR
J. Hsiang, D. F. Hsu and Y. P. Shieh (arping(AT)turing.csie.ntu.edu.tw), Jun 03 2002
EXTENSIONS
Offset corrected by Sean A. Irvine, Aug 04 2024
STATUS
approved