A176127 The number of permutations of {1,2,...,n,1,2,...,n} with the property that there are k numbers between the two k's in the set for k=1,...,n.
0, 0, 2, 2, 0, 0, 52, 300, 0, 0, 35584, 216288, 0, 0, 79619280, 653443600, 0, 0, 513629782560, 5272675722400, 0, 0, 7598911885030976, 93690316113031872, 0, 0, 223367222197529806464, 3214766521218764786304, 0, 0
Offset: 1
Examples
a(1)=0; a(2)=0; a(3)=a(4)=2 since {{2,3,1,2,1,3},{3,1,2,1,3,2}} and {{4,1,3,1,2,4,3,2},{2,3,4,2,1,3,1,4}} are the only ways to permute {1,2,3,1,2,3} and {1,2,3,4,1,2,3,4}, respectively, such that there is one number between the 1's, two numbers between the 2's,..., n numbers between the n's.
References
- For references, see A014552.
Links
- Ali Assarpour, Amotz Bar-Noy, Ou Liuo, Counting the Number of Langford Skolem Pairings, arXiv:1507.00315 [cs.DM], 2015.
Programs
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Sage
a=lambda n:sum(1 for i in DLXCPP([(i-1,j+n,i+j+n+1)for i in[1..n]for j in[0..n+n-i-2]]+[(i,)for i in[n..n+n-1]]))if n%4 in[0,3] else 0 # Tomas Boothby, Jun 14 2013
Formula
a(n) = 2 * A014552(n).
Extensions
Edited and more terms added from A014552 by Max Alekseyev, May 31 2011, May 19 2015
Corrected and extended using results from the Assarpour et al. (2015) paper by N. J. A. Sloane, Feb 22 2016 at the suggestion of William Rex Marshall.