A071608 -id:A071608 - cp's OEIS frontend

A003111 Number of complete mappings of the cyclic group Z_{2n+1}.

1, 1, 3, 19, 225, 3441, 79259, 2424195, 94471089, 4613520889, 275148653115, 19686730313955, 1664382756757625

Offset: 0

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

			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.

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).

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, and 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, and 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 and I. M. Wanless, A congruence connecting Latin rectangles and partial orthomorphisms, Ann. Comb. 16, No. 2, 349-365 (2012).

Bisection of A006609.

Cf. A006717, A071607, A071608, A071706, A006204, A003109, A003110.

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]. - Ian Wanless, Jul 30 2010
a(n) = A003109(n) + A003110(n). - Sean A. Irvine, Jan 30 2015
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)

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

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

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

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.

			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.

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).

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.

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

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

cp's OEIS Frontend

A003111 Number of complete mappings of the cyclic group Z_{2n+1}.

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