A272363 -id:A272363 - cp's OEIS frontend

A320129 Number of ways to group the first 2*n natural numbers into n pairs (xi, yi) such that the n numbers xi + yi are all different.

1, 1, 2, 10, 55, 412, 3736, 40518, 505486, 7145031, 112844566, 1970286922, 37676184205

Offset: 0

Altug Alkan, Oct 06 2018

			For n = 2, a(2) = 2 since {(1,3), (2,4)} and {(1,2), (3,4)} are corresponding sets. {(2,4), (1,3)} and {(3,1), (4,2)} are considered to be the same grouping with {(1,3), (2,4)}.

Altug Alkan, Line plot of t(n)-t(n-1) where t(n) = a(n+1)/a(n) for n <= 11

Cf. A002968, A060963, A272363.

okperm(vp, n) = {for (k=1, n-1, if (vp[k] > vp[k+1], return (0)); ); for (k=1, n, if (vp[k+n] <= vp[k], return (0)); ); 1; }
a(n) = if(n==0, 1, {nb = 0; nn = 2*n; for (j=0, nn!-1, vp = numtoperm(nn, j); if (okperm(vp, n), vs = vector(n, k, vp[k]+vp[k+n]); if (#vs == #Set(concat(vs)), nb++); ); ); nb; } ) \\ after Michel Marcus at A272363

from sympy.utilities.iterables import multiset_partitions
def A320129(n):
    return 1 if n == 0 else sum(1 for p in multiset_partitions(list(range(1,2*n+1)),n) if max(len(d) for d in p) == 2 and len(set(sum(d) for d in p)) == n) # Chai Wah Wu, Oct 08 2018

a(n) >= A272363(n).

a(11)-a(12) from Giovanni Resta, Oct 09 2018

A007631 Number of solutions to non-attacking reflecting queens problem.

Original entry on oeis.org

1, 1, 0, 0, 2, 4, 0, 2, 10, 32, 38, 140, 496, 1186, 3178, 16792, 82038, 289566, 1139874, 5914118, 33800010, 142337180, 721286448, 4384569864

Offset: 0

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

a(n) is the number of ways to pair the natural numbers from 1 to n with those between n+1 and 2*n into n pairs (xi,yi) such that the 2*n numbers yi+i and yi-i are all different. - Michel Marcus, Apr 27 2016

			For n = 4, ((1,7), (2,5), (3,8), (4,6)) is an instance of such grouping. ((2,5), (1,7), (3,8), (4,6)) is considered to be the same grouping.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Jordan Bell, Brett Stevens, A survey of known results and research areas for n-queens, Discrete Mathematics, Volume 309 (2009), pp 1-31.
M. Gardner, Fractal Music, Hypercards and More, Freeman, NY, 1991, p. 240.
G. B. Huff, On pairings of the first 2n natural numbers, Acta. Arith. 23 (1973) 117-126.
D. A. Klarner, The Problem of Reflecting Queens, The American Mathematical Monthly, Vol. 74, No. 8 (Oct., 1967), pp. 953-955.
M. Slater, Number theory Research Problem 1, Bull. Amer. Math. Soc. 69 (1963), 333.

Cf. A000170, A002968, A051223, A051224, A272363.

a(n) = {nb = 0; for (j=0, n!-1, vp = numtoperm(n, j); vb = vector(n, k, vp[k]+n); vs = vector(n, k, vb[k]+k); vd = vector(n, k, vb[k]-k); if (#vs + #vd == #Set(concat(vs, vd)), nb++); ); nb; } \\ Michel Marcus,  Apr 27 2016

a(18)-a(21) from Sean A. Irvine, Jan 13 2018
a(0)-a(3) prepended by Michel Marcus, Oct 03 2018
a(22) from Sean A. Irvine, Oct 04 2018
a(23) from Sean A. Irvine, Oct 07 2018

A320168 Number of ways to group the first 2*n positive integers into n pairs (xi, yi) with xi < yi, and such that the n numbers (yi mod xi) are all different.

Original entry on oeis.org

1, 1, 2, 2, 7, 12, 22, 26, 85, 226, 717, 1695, 5071, 14275, 47405, 176747, 638329, 2166516

Offset: 0

Altug Alkan, Oct 07 2018

How does a(n+1)/a(n) behave as n increases?

			a(3) = 2 because {(1,3), (2,5), (4,6)} and {(1,5), (2,3), (4,6)} are corresponding sets.
a(4) = 7 because {(1,6), (2,5), (3,8), (4,7)}, {(1,3), (2,7), (4,6), (5,8)}, {(1,7), (2,3), (4,6), (5,8)}, {(1,3), (2,5), (4,7), (6,8)}, {(1,5), (2,3), (4,7), (6,8)}, {(1,2), (3,7), (4,6), (5,8)}, {(1,2), (3,8), (4,7), (5,6)} are corresponding sets.

David A. Corneth, Illustration for a(4)

Cf. A002968, A060963, A272363, A320129.

a(13)-a(17) from Rémy Sigrist, Oct 07 2018

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A320129 Number of ways to group the first 2*n natural numbers into n pairs (xi, yi) such that the n numbers xi + yi are all different.

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A007631 Number of solutions to non-attacking reflecting queens problem.

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A320168 Number of ways to group the first 2*n positive integers into n pairs (xi, yi) with xi < yi, and such that the n numbers (yi mod xi) are all different.

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Examples

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