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A322442
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.
10
1, 1, 4, 25, 195, 1894, 22159, 303769, 4790858, 85715595, 1720097275, 38355019080, 942872934661, 25383601383937, 744118939661444, 23635548141900445, 809893084668253151, 29822472337116844174, 1175990509568611058299, 49504723853840395163221, 2218388253903492656783562
OFFSET
0,3
LINKS
FORMULA
E.g.f.: exp(exp(x)-1) * (2*B(x) - 1) where B(x) is the e.g.f. of A319884. - Andrew Howroyd, Jan 19 2024
EXAMPLE
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)
MATHEMATICA
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}]
PROG
(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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 08 2018
EXTENSIONS
a(8) onwards from Andrew Howroyd, Jan 19 2024
STATUS
approved