A318131 Number of non-isomorphic sets of finite (possibly empty) sets with union {1,2,...,n} and intersection {}.

1, 1, 6, 60, 3836, 37325360, 25626412263611792, 67516342973185974276922865448446208, 2871827610052485009904013737758920847534777143951264797898686184985092096

Offset: 0

Author

Gus Wiseman, Aug 18 2018

Keywords

• nonn

Examples

			Non-isomorphic representatives of the a(2) = 6 sets of sets:
  {{1},{2}}
  {{},{1,2}}
  {{},{1},{2}}
  {{},{1},{1,2}}
  {{1},{2},{1,2}}
  {{},{1},{2},{1,2}}
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..12

Crossrefs

Cf. A000371, A000612, A003465, A055621, A119563, A131288, A283877, A293606, A304997.

Cf. A318128, A318129, A318130, A318132.

Programs

Mathematica

sysnorm[m_]:=If[Union@@m!=Range[Max@@Flatten[m]],sysnorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[sysnorm[m,1]]]];sysnorm[m_,aft_]:=If[Length[Union@@m]<=aft,{m},With[{mx=Table[Count[m,i,{2}],{i,Select[Union@@m,#>=aft&]}]},Union@@(sysnorm[#,aft+1]&/@Union[Table[Map[Sort,m/.{par+aft-1->aft,aft->par+aft-1},{0,1}],{par,First/@Position[mx,Max[mx]]}]])]];
Table[Length[Union[sysnorm/@Select[Subsets[Subsets[Range[n]]],And[Union@@#===Range[n],Intersection@@#=={}]&]]],{n,4}]

Formula

a(n) = 2*(A055621(n) - A055621(n-1)) = 2*(A000612(n) - 2*A000612(n-1) + A000612(n-2)) for n >= 2. - Andrew Howroyd, Jan 29 2024

Extensions

a(5) onwards from Andrew Howroyd, Jan 29 2024