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A319884
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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.
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3
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1, 0, 1, 7, 50, 481, 5667, 78058, 1238295, 22314627, 451354476, 10148011215, 251584513215, 6831141750512, 201976943666357, 6470392653260939, 223595676728884394, 8302299221314559877, 330075531021130110015, 14006780163088113914026, 632606447496264724088803
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OFFSET
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0,4
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LINKS
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FORMULA
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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
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EXAMPLE
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The a(3) = 7 pairs of set partitions:
(1)(2)(3)|(123)
(1)(23)|(12)(3)
(1)(23)|(13)(2)
(1)(23)|(123)
(12)(3)|(13)(2)
(12)(3)|(123)
(13)(2)|(123)
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MATHEMATICA
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sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
costabstrQ[s_, t_]:=And@@Cases[s, x_:>Select[t, x!=#&&(SubsetQ[x, #]||SubsetQ[#, x])&]!={}];
Table[Length[Select[Subsets[sps[Range[n]], {2}], And[costabstrQ@@#, costabstrQ@@Reverse[#]]&]], {n, 5}]
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PROG
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(PARI)
F(x)={my(bell=(exp(y*(exp(x) - 1)) )); subst(serlaplace( serconvol(bell, bell)), y, exp(exp(x) - 1)-1)}
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
(PARI) \\ 2nd prog, following formula - slightly slower
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))
seq(n) = Vec(serlaplace((1/2)*(1 + D(n, exp(exp(x + O(x*x^n)) - 1) - 1)))) \\ Andrew Howroyd, Jan 20 2024
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CROSSREFS
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Cf. A000110, A000258, A001247, A008277, A059849, A060639, A181939, A322435, A322436, A322437, A322438, A322439, A322440, A322441, A322442.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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