login
A318131
Number of non-isomorphic sets of finite (possibly empty) sets with union {1,2,...,n} and intersection {}.
5
1, 1, 6, 60, 3836, 37325360, 25626412263611792, 67516342973185974276922865448446208, 2871827610052485009904013737758920847534777143951264797898686184985092096
OFFSET
0,3
LINKS
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
EXAMPLE
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}}
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}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 18 2018
EXTENSIONS
a(5) onwards from Andrew Howroyd, Jan 29 2024
STATUS
approved