0,3

A complete mapping of a cyclic group (Z_n,+) is a permutation f(x) of Z_n such that f(0)=0 and such that f(x)-x is also a permutation.

a(n)=TSQ(n)/n where TSQ(n) is the number of solutions of the toroidal semi-n-queen problem (A006717 is the sequence TSQ(2k-1)).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Anthony B. Evans, Orthomorphism Graphs of Groups, vol. 1535 of Lecture Notes in Mathematics, Springer-Verlag, 1991.

J. Hsiang, D. F. Hsu and Y. P. Shieh, On the hardness of counting problems of complete mappings, Discrete Math., 277 (2004), 87-100.

Lehmer, D. H.; Some properties of circulants. J. Number Theory 5 (1973), 43-54.

B. D. McKay, J. C. McLeod and I. M. Wanless, The number of transversals in a Latin square, Des. Codes Cryptogr., 40, (2006) 269-284.

D. Novakovic, (2000) Computation of the number of complete mappings for permutations. Cybernetics & System Analysis, No. 2, v. 36, pp. 244-247.

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.

Y. P. Shieh, Cyclic complete mappings counting problems

Every term is odd and if n=2 mod 3 then a(n) is divisible by 3. Also a(n) is asymptotically less than 0.62^n n!. [McKay, McLeod, Wanless]

f(x)=2x in (Z_7,+) is a complete mapping of Z_7 since that f(0)=0 and that f(x)-x (=x) is also a permutation of Z_7.

Cf. A006717, A071607, A071608, A071706, A006204.

Adjacent sequences: A003108 A003109 A003110 this_sequence A003112 A003113 A003114

Sequence in context: A166380 A136652 A136504 this_sequence A126444 A001929 A157675

nonn,nice

N. J. A. Sloane (njas(AT)research.att.com).

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

a(12) from Yuh-Pyng Shieh (arping(AT)gmail.com), Jan 10 2006