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.

%I A320129 #37 Jan 14 2024 12:11:00
%S A320129 1,1,2,10,55,412,3736,40518,505486,7145031,112844566,1970286922,
%T A320129 37676184205
%N 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.
%H A320129 Altug Alkan, <a href="/A320129/a320129.png">Line plot of t(n)-t(n-1) where t(n) = a(n+1)/a(n) for n <= 11</a>
%F A320129 a(n) >= A272363(n).
%e A320129 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)}.
%o A320129 (PARI)
%o A320129 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; }
%o A320129 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
%o A320129 (Python)
%o A320129 from sympy.utilities.iterables import multiset_partitions
%o A320129 def A320129(n):
%o A320129     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
%Y A320129 Cf. A002968, A060963, A272363.
%K A320129 nonn,more
%O A320129 0,3
%A A320129 _Altug Alkan_, Oct 06 2018
%E A320129 a(11)-a(12) from _Giovanni Resta_, Oct 09 2018