OFFSET
0,3
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..200
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
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 08 2018
EXTENSIONS
a(8) onwards from Andrew Howroyd, Jan 19 2024
STATUS
approved