A317794 Number of non-isomorphic set-systems on n vertices with no singletons.

1, 1, 2, 8, 180, 612032, 200253854316544, 263735716028826427534807159537664, 5609038300883759793482640992086670066760184863720423808367168537493504

Offset

0,3

Links

Table of n, a(n) for n=0..8.

Loïc Foissy, Hopf algebraic structures on hypergraphs and multi-complexes, arXiv:2304.00810 [math.CO], 2023.

Example

Non-isomorphic representatives of the a(3) = 8 set-systems:
  0,
  {12}, {123},
  {12}{13}, {12}{123},
  {12}{13}{23}, {12}{13}{123},
  {12}{13}{23}{123}.

Mathematica

sysnorm[{}] := {}; 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[Select[Subsets[Range[n]], Length[#]>1&]], Or[Length[#]==0, Union@@#==Range[Max@@Union@@#]]&]]], {n, 4}]
(* second program *)
Table[Sum[2^PermutationCycles[Ordering[Map[Sort, Subsets[Range[n], {2, n}]/.Rule@@@Table[{i, prm[[i]]}, {i, n}], {1}]], Length]/n!, {prm, Permutations[Range[n]]}], {n, 6}] (* Gus Wiseman, Dec 12 2018 *)

Crossrefs

The spanning case is A317795.

Cf. A000088, A000612, A003180, A007716, A055621, A283877, A300913, A306005, A317533, A317757, A319876.

Sequence in context: A009713 A265597 A213200 * A009377 A181234 A156526

Adjacent sequences: A317791 A317792 A317793 * A317795 A317796 A317797

Keyword

nonn

Author

Gus Wiseman, Aug 07 2018

Extensions

More terms from Gus Wiseman, Dec 12 2018

Status

approved