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Search: id:A006204
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| A006204 |
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Number of starters in cyclic group of order 2n+1.
(Formerly M2802)
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+0
2
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| 1, 1, 3, 9, 25, 133, 631, 3857, 25905, 188181, 1515283, 13376125, 128102625, 1317606101, 14534145947, 170922533545, 2138089212789
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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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.
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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.
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.
Classification of starters, V. Linja-aho and P. R. J. Ostergard, Journal of Combinatorial Mathematics and Combinatorial Computing, to appear. [From Vesa Linja-aho (vesa.linja-aho(AT)tkk.fi), May 06 2009]
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LINKS
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Y. P. Shieh, Cyclic complete mappings counting problems
Classification of starters, V. Linja-aho and P. R. J. Ostergard, Journal of Combinatorial Mathematics and Combinatorial Computing, to appear. [From Vesa Linja-aho (vesa.linja-aho(AT)tkk.fi), May 06 2009]
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EXAMPLE
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f(x)=6x in (Z_7,+) is a complete mapping of Z_7 since that f(0)=0 and that f(x)-x (=5x) is also a permutation of Z_7. f^(-1)(x)=6x=f(x). So f(x) is fixed under reflection.
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CROSSREFS
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Cf. A006717, A071607, A071608, A071706, A003111.
Sequence in context: A012771 A120284 A074440 this_sequence A013572 A119851 A119825
Adjacent sequences: A006201 A006202 A006203 this_sequence A006205 A006206 A006207
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KEYWORD
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nonn,nice,hard,more
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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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 (Roland.Bacher(AT)ujf-grenoble.fr), Dec 18 2007
Extended by Vesa Linja-aho (vesa.linja-aho(AT)tkk.fi), May 06 2009
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