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%I #37 Feb 10 2024 15:41:38
%S 1,0,0,1,1,3,4,7,9,16,21,33,46,68,95,140,187,266,372,507,683,948,1256,
%T 1692,2263,3003,3955,5248,6824,8921,11669,15058,19413,25128,32149,
%U 41129,52578,66740,84696,107389,135310,170277,214386,268151,335261,418896,521204
%N Number of unordered pairs of partitions of n (into distinct parts) with empty intersection.
%C Counted as orderless pairs since intersection is commutative.
%H Vaclav Kotesovec <a href="/A108796/b108796.txt">Table of n, a(n) for n = 0..1050</a> (terms 0..700 from Alois P. Heinz)
%F a(n) = ceiling(1/2 * [(x*y)^n] Product_{j>0} (1+x^j+y^j)). - _Alois P. Heinz_, Mar 31 2017
%F a(n) = ceiling(A365662(n)/2). - _Gus Wiseman_, Oct 07 2023
%e Of the partitions of 12 into different parts, the partition (5+4+2+1) has an empty intersection with only (12) and (9+3).
%e From _Gus Wiseman_, Oct 07 2023: (Start)
%e The a(6) = 4 pairs are:
%e ((6),(5,1))
%e ((6),(4,2))
%e ((6),(3,2,1))
%e ((5,1),(4,2))
%e (End)
%t using DiscreteMath`Combinatorica`and ListPartitionsQ[n_Integer]:= Flatten[ Reverse /@ Table[(Range[m-1, 0, -1]+#1&)/@ TransposePartition/@ Complement[Partitions[ n-m* (m-1)/2, m], Partitions[n-m*(m-1)/2, m-1]], {m, -1+Floor[1/2*(1+Sqrt[1+8*n])]}], 1]; Table[Plus@@Flatten[Outer[If[Intersection[Flatten[ #1], Flatten[ #2]]==={}, 1, 0]&, ListPartitionsQ[k], ListPartitionsQ[k], 1]], {k, 48}]/2
%t nmax = 50; p = 1; Do[p = Expand[p*(1 + x^j + y^j)]; p = Select[p, (Exponent[#, x] <= nmax) && (Exponent[#, y] <= nmax) &], {j, 1, nmax}]; p = Select[p, Exponent[#, x] == Exponent[#, y] &]; Table[Coefficient[p, x^n*y^n]/2, {n, 1, nmax}] (* _Vaclav Kotesovec_, Apr 07 2017 *)
%t Table[Length[Select[Subsets[Select[IntegerPartitions[n], UnsameQ@@#&],{2}],Intersection@@#=={}&]],{n,15}] (* _Gus Wiseman_, Oct 07 2023 *)
%o (PARI) a(n) = {my(A=1 + O(x*x^n) + O(y*y^n)); polcoef(polcoef(prod(k=1, n, A + x^k + y^k), n), n)/2} \\ _Andrew Howroyd_, Oct 10 2023
%Y Column k=2 of A258280.
%Y Cf. A079122, A086737.
%Y Main diagonal of A284593 times (1/2).
%Y This is the strict case of A260669.
%Y The ordered version is A365662 = strict case of A054440.
%Y This is the disjoint case of A366132, with twins A366317.
%Y A000041 counts integer partitions, strict A000009.
%Y A002219 counts biquanimous partitions, strict A237258, ordered A064914.
%Y Cf. A000124, A000712, A001255, A006827, A032302, A122768, A355389.
%K nonn
%O 0,6
%A _Wouter Meeussen_, Jul 09 2005
%E Name edited by _Gus Wiseman_, Oct 10 2023
%E a(0)=1 prepended by _Alois P. Heinz_, Feb 09 2024
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