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%I #36 Jan 20 2024 11:08:27
%S 1,0,1,7,50,481,5667,78058,1238295,22314627,451354476,10148011215,
%T 251584513215,6831141750512,201976943666357,6470392653260939,
%U 223595676728884394,8302299221314559877,330075531021130110015,14006780163088113914026,632606447496264724088803
%N Number of unordered pairs of set partitions of {1,...,n} where every block of one is a proper subset or proper superset of some block of the other.
%H Andrew Howroyd, <a href="/A319884/b319884.txt">Table of n, a(n) for n = 0..200</a>
%F E.g.f.: (1/2)*(1 + D(x, exp(exp(x) - 1) - 1) ) where D(x,y) = (exp(2*y)/(1 + y)^2) * Sum_{k>=0} x^k*(Sum_{j=0..k} Stirling2(k,j)*y^j)^2/k!. - _Andrew Howroyd_, Jan 20 2024
%e The a(3) = 7 pairs of set partitions:
%e (1)(2)(3)|(123)
%e (1)(23)|(12)(3)
%e (1)(23)|(13)(2)
%e (1)(23)|(123)
%e (12)(3)|(13)(2)
%e (12)(3)|(123)
%e (13)(2)|(123)
%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t costabstrQ[s_,t_]:=And@@Cases[s,x_:>Select[t,x!=#&&(SubsetQ[x,#]||SubsetQ[#,x])&]!={}];
%t Table[Length[Select[Subsets[sps[Range[n]],{2}],And[costabstrQ@@#,costabstrQ@@Reverse[#]]&]],{n,5}]
%o (PARI)
%o F(x)={my(bell=(exp(y*(exp(x) - 1)) )); subst(serlaplace( serconvol(bell, bell)), y, exp(exp(x) - 1)-1)}
%o seq(n) = {my(x=x + O(x*x^n)); Vec(serlaplace( 1 + exp( 2*(exp(exp(x) - 1) - exp(x)) ) * F(x) )/2)} \\ _Andrew Howroyd_, Jan 19 2024
%o (PARI) \\ 2nd prog, following formula - slightly slower
%o D(n,y) = (exp(2*y)/(1 + y)^2) * sum(k=0,n, x^k*sum(j=0, k, stirling(k,j,2) * y^j)^2/k!, O(x*x^n))
%o seq(n) = Vec(serlaplace((1/2)*(1 + D(n, exp(exp(x + O(x*x^n)) - 1) - 1)))) \\ _Andrew Howroyd_, Jan 20 2024
%Y Cf. A000110, A000258, A001247, A008277, A059849, A060639, A181939, A322435, A322436, A322437, A322438, A322439, A322440, A322441, A322442.
%K nonn
%O 0,4
%A _Gus Wiseman_, Dec 09 2018
%E a(8) onwards from _Andrew Howroyd_, Jan 19 2024
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