A264813 -id:A264813 - cp's OEIS frontend

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

Andrew McFarland, Apr 09 2010

			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.

For references, see A014552.

Ali Assarpour, Amotz Bar-Noy, Ou Liuo, Counting the Number of Langford Skolem Pairings, arXiv:1507.00315 [cs.DM], 2015.

Cf. A014552, A059106, A004075, A264813, A268536.

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

a(n) = 2 * A014552(n).

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.

A203435 Number of partitions of {1,2,...,4n} into n 4-element subsets having the same sum.

Original entry on oeis.org

1, 1, 4, 32, 392, 6883, 171088, 5661874, 242038179, 13147317481

Offset: 0

Alois P. Heinz, Jan 01 2012

The element sum of each subset is 8n+2. The larger terms were computed with Knuth's dancing links algorithm.

			a(1) = 1: {1,2,3,4}.
a(2) = 4: {1,2,7,8}, {3,4,5,6}; {1,3,6,8}, {2,4,5,7}; {1,4,5,8}, {2,3,6,7}; {1,4,6,7}, {2,3,5,8}.

Wikipedia, Dancing Links
Wikipedia, Partition of a set

Column k=4 of A203986.

Cf. A104185, A108235, A203017, A264813.

b:= proc() option remember; local i, j, t, m; m:= args[nargs]; if args[1]=0 then `if`(nargs=2, 1, b(args[t] $t=2..nargs)) elif args[1]<1 then 0 else add(`if`(args[j] `if`(n=0, 1, b(((8*n+2)+4/97) $n, 4*n)/n!): seq(a(n), n=0..6);

b[l_] := b[l] = Module[{nl = Length[l], k = l[[-1]], m = l[[-2]]}, Which[l[[1]] == 0, If[nl == 3, 1, b[l[[2 ;; nl]]]], l[[1]] < 1, 0, True, Sum[If[l[[j]] < m, 0, b[Join[Sort[Table[l[[i]] - If[i == j, m + 1/97, 0], {i, 1, nl - 2}]], {m - 1, k}]]], {j, 1, nl - 2}]]];
a[n_] := If[n == 0, 1, b[Join[Array[8*n + 2 + 4/97& , n], {4*n, 4}]]/n!];
Table[a[n], {n, 0, 6}] (* Jean-François Alcover, Jun 03 2018, adapted from Maple *)

A321956 Number of permutations of 3 indistinguishable copies of 1,...,n such that the first and second copies of j are adjacent and there are exactly j-1 numbers between the second and the third copy of j.

Original entry on oeis.org

1, 1, 0, 2, 4, 0, 16, 40, 0, 456, 1759, 0, 34636, 175198, 0, 5494621, 34043062, 0

Offset: 0

Alois P. Heinz, Nov 22 2018

			a(3) = 2: 111332232, 332232111.
a(4) = 4: 111334432242, 332232441114, 334432242111, 441114332232.

Eric Weisstein's World of Mathematics, Langford's Problem
Wikipedia, Dancing Links
Wikipedia, Langford pairing

Cf. A264813, A322153.

a(n) = 0 for n == 2 (mod 3).

cp's OEIS Frontend

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.

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A203435 Number of partitions of {1,2,...,4n} into n 4-element subsets having the same sum.

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A321956 Number of permutations of 3 indistinguishable copies of 1,...,n such that the first and second copies of j are adjacent and there are exactly j-1 numbers between the second and the third copy of j.

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