A071706 Number of complete mappings f(x) of the cyclic group Z_{2n+1} such that -f(-x)=f.

1, 1, 3, 5, 21, 69, 319, 1957, 12513, 85445, 656771, 5591277, 51531405, 509874417, 5438826975, 62000480093, 752464463029, 9685138399785, 131777883431119

Offset: 0

Author

J. Hsiang, D. F. Hsu and Y. P. Shieh (arping(AT)turing.csie.ntu.edu.tw), Jun 03 2002

Keywords

• nonn

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.

Examples

			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.
		

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.

Links

• Y. P. Shieh, Cyclic complete mappings counting problems

Extensions

Offset corrected by Sean A. Irvine, Aug 04 2024