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.
1, 1, 4, 25, 195, 1894, 22159, 303769, 4790858, 85715595, 1720097275, 38355019080, 942872934661, 25383601383937, 744118939661444, 23635548141900445, 809893084668253151, 29822472337116844174, 1175990509568611058299, 49504723853840395163221, 2218388253903492656783562
Offset: 0
Keywords
Examples
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)
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..200
Crossrefs
Programs
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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}] -
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
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
Extensions
a(8) onwards from Andrew Howroyd, Jan 19 2024