OFFSET
0,3
COMMENTS
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)).
Stated another way, this is the number of "good" permutations on 2n+1 elements (see A006717) that start with 0. [Novakovich]. - N. J. A. Sloane, Feb 22 2011
REFERENCES
Anthony B. Evans, Orthomorphism Graphs of Groups, vol. 1535 of Lecture Notes in Mathematics, Springer-Verlag, 1991.
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
N. J. Cavenagh and I. M. Wanless, On the number of transversals in Cayley tables of cyclic groups, Disc. Appl. Math. 158 (2010), 136-146.
Sean Eberhard, F. Manners, R. Mrazovic, Additive triples of bijections, or the toroidal semiqueens problem, arXiv preprint arXiv:1510.05987 [math.CO], 2015-2016.
Jieh Hsiang, YuhPyng Shieh, and YaoChiang Chen, Cyclic Complete Mappings Counting Problems, National Taiwan University 2014/8/21.
J. Hsiang, D. F. Hsu and Y. P. Shieh, On the hardness of counting problems of complete mappings, Discrete Math., 277 (2004), 87-100.
N. Yu. Kuznetsov, Using the Monte Carlo Method for Fast Simulation of the Number of "Good" Permutations on the SCIT-4 Multiprocessor Computer Complex, Cybernetics and Systems Analysis, January 2016, Volume 52, Issue 1, pp 52-57.
D. H. Lehmer, 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, Computation of the number of complete mappings for permutations, Cybernetics & System Analysis, No. 2, v. 36 (2000), pp. 244-247.
S. V. S. Ranganathan, D. Divsalar, R. D. Wesel, On the Girth of (3, L) Quasi-Cyclic LDPC Codes based on Complete Protographs, arXiv preprint arXiv:1504.04975 [cs.IT], 2015.
Y. P. Shieh, Cyclic complete mappings counting problems
D. S. Stones and I. M. Wanless, Compound orthomorphisms of the cyclic group, Finite Fields Appl. 16 (2010), 277-289.
D. S. Stones, I. M. Wanless, A congruence connecting Latin rectangles and partial orthomorphisms, Ann. Comb. 16, No. 2, 349-365 (2012).
FORMULA
Suppose n is odd and let b(n)=a((n-1)/2). Then b(n) is odd; if n>3 and n is not 1 mod 3 then b(n) is divisible by 3; b(n)=-2 mod n in n is prime; b(n) is divisible by n if n is composite; b(n) is asymptotically in between 3.2^n and 0.62^n n!. [Cavenagh, Wanless], [McKay, McLeod, Wanless], [Stones, Wanless]. - From Ian Wanless, Jul 30 2010
Conjecture: a(n) = A006609(2*n+2), n>0. - Sean A. Irvine, Jan 30 2015
From Vaclav Kotesovec, Jul 22 2023: (Start)
a(n) ~ exp(-1/2) * (2*n)!^2 / (2*n + 1)^(2*n - 1). [Eberhard, Manners, Mrazovic, 2016, Theorem 1.3, n->2*n+1]
a(n) ~ Pi * 2^(2*n + 3) * n^(2*n + 2) / exp(4*n + 3/2). (End)
EXAMPLE
f(x)=2x in (Z_7,+) is a complete mapping of Z_7 since f(0)=0 and f(x)-x (=x) is also a permutation of Z_7.
CROSSREFS
KEYWORD
nonn,nice,more
AUTHOR
EXTENSIONS
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
STATUS
approved