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A006717 Number of ways of arranging 2n+1 nonattacking semi-queens on a (2n+1) X (2n+1) toroidal board.
(Formerly M3005)
1, 3, 15, 133, 2025, 37851, 1030367, 36362925, 1606008513, 87656896891, 5778121715415, 452794797220965, 41609568918940625 (list; graph; refs; listen; history; text; internal format)



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


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.


Table of n, a(n) for n=0..12.

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.

V. 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.


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 n!/e [Eberhard, Manners, Mrazovic]. - Sam Spiro, Apr 16 2019

a(n) = (2*n+1) * A003111(n). - Andrew Howroyd, Sep 28 2020


(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


Cf. A003111, A007705.

Sequence in context: A222390 A281186 A108210 * A222263 A246804 A230166

Adjacent sequences:  A006714 A006715 A006716 * A006718 A006719 A006720




N. J. A. Sloane


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



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Last modified January 23 03:02 EST 2021. Contains 340384 sequences. (Running on oeis4.)