A006204 Number of starters in cyclic group of order 2n+1.

1, 1, 3, 9, 25, 133, 631, 3857, 25905, 188181, 1515283, 13376125, 128102625, 1317606101, 14534145947, 170922533545, 2138089212789

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• hard
• more

Comments

A complete mapping of a cyclic group (Z_m,+) is a permutation f(x) of Z_m with f(0)=0 such that f(x)-x is also a permutation. a(n) is the number of complete mappings f(x) of the cyclic group Z_{2n+1} such that f^(-1)=f.

In other words, a(n) is the number of complete mappings fixed under the reflection operator R, where R(f)=f^(-1). Reflection R is not only a symmetry operator of complete mappings, but also one of the (Toroidal)-(semi) N-Queen problems and of the strong complete mappings problem.

Examples

			f(x)=6x in (Z_7,+) is a complete mapping of Z_7 since f(0)=0 and f(x)-x (=5x) is also a permutation of Z_7. f^(-1)(x)=6x=f(x). So f(x) is fixed under reflection.
		

References

• CRC Handbook of Combinatorial Designs, 1996, p. 469.
• CRC Handbook of Combinatorial Designs, 2nd edition, 2007, p. 624.
• J. D. Horton, Orthogonal starters in finite Abelian groups, Discrete Math., 79 (1989/1990), 265-278.
• V. Linja-aho and Patric R. J. Östergård, Classification of starters, J. Combin. Math. Combin. Comput. 75 (2010), 153-159.
• 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.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Bill Butler, Durango Bill's Bridge Probabilities and Combinatorics
• Jieh Hsiang, Yuhpyng Shieh, Yaochiang Chen, Cyclic complete mappings counting problems, National Taiwan University, Taipei, April 2003.
• Vesa Linja-aho, Patric R. J. Östergård, Classification of starters, J. Combin. Math. Combin. Comput. 75 (2010), 153-159.

Crossrefs

Cf. A006717, A071607, A071608, A071706, A003111.

Extensions

Additional comments and one more term from J. Hsiang, D. F. Hsu and Y. P. Shieh (arping(AT)turing.csie.ntu.edu.tw), Jun 03 2002

Corrected and extended by Roland Bacher, Dec 18 2007

Extended by Vesa Linja-aho (vesa.linja-aho(AT)tkk.fi), May 06 2009