OFFSET
0,2
COMMENTS
Also the number of "good" permutations on 2n+1 elements [Novakovich]. - N. J. A. Sloane, Feb 22 2011
Also the number of transversals of a cyclic Latin square of order 2n+1 and the number of orthomorphisms of the cyclic group of order 2n+1. - Ian Wanless, Oct 07 2001
Also the number of complete mappings of a cyclic group of order 2n+1; also (2n+1) times the number of "standard" complete mappings of cyclic group of order 2n+1. - Jieh Hsiang, D. Frank Hsu and Yuh Pyng Shieh (arping(AT)turing.csie.ntu.edu.tw), May 08 2002
See A003111 for further information.
A very simple model using only addition mod n: Let i=index vector (0,1,..n-1) on any set of n distinct values, and j=index vector for the values after reordering. Then j = (i + d) mod n, where d is the vector of distances moved, and a(n) = number of reorderings that give an equidistributed set d (i.e., 1 instance of each distance moved). Since a(n)=0 for all even n, taking only odd n gives the sequence above - Ross Drewe, Sep 03 2017
All broken diagonals and antidiagonals of cyclic Latin square are transversals, so a(n) >= 2*n for all n > 1 for which cyclic Latin squares exist. - Eduard I. Vatutin, Mar 23 2022
REFERENCES
Yuh Pyng Shieh, Jieh Hsiang and D. Frank Hsu, On the enumeration of Abelian k-complete mappings, vol. 144 of Congressus Numerantium, 2000, pp. 67-88.
Yuh Pyng Shieh, Partition Strategies for #P-complete problem with applications to enumerative combinatorics, PhD thesis, National Taiwan University, 2001.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
I. Vardi, Computational Recreations in Mathematica. Addison-Wesley, Redwood City, CA, 1991, p. 118.
LINKS
Christian Carley, The Name Tag Problem, Mathematics Undergraduate Theses (Boise State University, 2019).
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.
S. Eberhard, F. Manners, and R. Mrazovic, Additive Triples of Bijections, or the Toroidal Semiqueens Problem , arxiv:1510.05987, [math.CO], 2016.
Jieh Hsiang, YuhPyng Shieh, and YaoChiang Chen, Cyclic Complete Mappings Counting Problems, National Taiwan University 2014/8/21.
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013.
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.
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.
Kevin Pratt, Closed-Form Expressions for the n-Queens Problem and Related Problems, arXiv:1609.09585 [cs.DM], 2016.
D. S. Stones and I. M. Wanless, Compound orthomorphisms of the cyclic group, Finite Fields Appl. 16 (2010), 277-289.
Eric Weisstein's World of Mathematics, Queens Problem.
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 3n; b(n)=-2n mod n^2 in n is prime; b(n) is divisible by n^2 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
b(n) is asymptotic to e^(-1/2) n!^2/n^(n-1) [Eberhard, Manners, Mrazovic]. - Sam Spiro, Apr 16 2019; corrected by Sean Eberhard, Jul 21 2023
a(n) = (2*n+1) * A003111(n). - Andrew Howroyd, Sep 28 2020
PROG
(MATLAB) k = 6; A = zeros(1, k); for i = 1:k; n = 2*i-1; x = [0: n-1]; allP = perms(x); T = size(allP, 1); X = repmat(x, T, 1); Y = mod(X + allP, n); Y = sort(Y, 2); L = ~(sum(Y ~= X, 2)); A(i) = sum(L); end; A
% 1st 6 terms by testing all n! possible distance vectors
% Ross Drewe, Sep 03 2017
CROSSREFS
KEYWORD
nonn,more,nice
AUTHOR
EXTENSIONS
More terms from Jieh Hsiang, D. Frank Hsu and Yuh Pyng Shieh (arping(AT)turing.csie.ntu.edu.tw), May 08 2002
a(12) added from A003111 by N. J. A. Sloane, Mar 29 2007
Definition clarified by Vaclav Kotesovec, Sep 16 2014
STATUS
approved