A319884 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.
1, 0, 1, 7, 50, 481, 5667, 78058, 1238295, 22314627, 451354476, 10148011215, 251584513215, 6831141750512, 201976943666357, 6470392653260939, 223595676728884394, 8302299221314559877, 330075531021130110015, 14006780163088113914026, 632606447496264724088803
Offset: 0
Keywords
Examples
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)
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,_}]; 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}] -
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
Formula
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
Extensions
a(8) onwards from Andrew Howroyd, Jan 19 2024