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A322442
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Number of pairs of set partitions of {1,...,n} where every block of one is a subset or superset of some block of the other.
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10
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1, 1, 4, 25, 195, 1894, 22159, 303769, 4790858, 85715595, 1720097275, 38355019080, 942872934661, 25383601383937, 744118939661444, 23635548141900445, 809893084668253151, 29822472337116844174, 1175990509568611058299, 49504723853840395163221, 2218388253903492656783562
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OFFSET
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0,3
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LINKS
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FORMULA
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EXAMPLE
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The a(3) = 25 pairs of set partitions (these are actually all pairs of set partitions of {1,2,3}):
(1)(2)(3)|(1)(2)(3)
(1)(2)(3)|(1)(23)
(1)(2)(3)|(12)(3)
(1)(2)(3)|(13)(2)
(1)(2)(3)|(123)
(1)(23)|(1)(2)(3)
(1)(23)|(1)(23)
(1)(23)|(12)(3)
(1)(23)|(13)(2)
(1)(23)|(123)
(12)(3)|(1)(2)(3)
(12)(3)|(1)(23)
(12)(3)|(12)(3)
(12)(3)|(13)(2)
(12)(3)|(123)
(13)(2)|(1)(2)(3)
(13)(2)|(1)(23)
(13)(2)|(12)(3)
(13)(2)|(13)(2)
(13)(2)|(123)
(123)|(1)(2)(3)
(123)|(1)(23)
(123)|(12)(3)
(123)|(13)(2)
(123)|(123)
Non-isomorphic representatives of the pairs of set partitions of {1,2,3,4} for which the condition fails:
(12)(34)|(13)(24)
(12)(34)|(1)(3)(24)
(1)(2)(34)|(13)(24)
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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, ___}];
costabQ[s_, t_]:=And@@Cases[s, x_:>Select[t, SubsetQ[x, #]||SubsetQ[#, x]&]!={}];
Table[Length[Select[Tuples[sps[Range[n]], 2], And[costabQ@@#, costabQ@@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( exp( 2*exp(exp(x) - 1) - exp(x) - 1) * F(x) ))} \\ Andrew Howroyd, Jan 19 2024
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CROSSREFS
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Cf. A000110, A000258, A001247, A008277, A059849, A060639, A181939, A318393, A322435, A322437, A322439, A322440, A322441.
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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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