A330229 Number of fully chiral set-systems covering n vertices.

1, 1, 2, 42, 21336

Offset: 0

Author

Gus Wiseman, Dec 08 2019

Keywords

• nonn
• more

Comments

A set-system is a finite set of finite nonempty sets. It is fully chiral if every permutation of the vertices gives a different representative.

Examples

			The a(3) = 42 set-systems:
  {1}{2}{13}    {1}{2}{12}{13}    {1}{2}{12}{13}{123}
  {1}{2}{23}    {1}{2}{12}{23}    {1}{2}{12}{23}{123}
  {1}{3}{12}    {1}{3}{12}{13}    {1}{3}{12}{13}{123}
  {1}{3}{23}    {1}{3}{13}{23}    {1}{3}{13}{23}{123}
  {2}{3}{12}    {2}{3}{12}{23}    {2}{3}{12}{23}{123}
  {2}{3}{13}    {2}{3}{13}{23}    {2}{3}{13}{23}{123}
  {1}{12}{23}   {1}{2}{13}{123}
  {1}{13}{23}   {1}{2}{23}{123}
  {2}{12}{13}   {1}{3}{12}{123}
  {2}{13}{23}   {1}{3}{23}{123}
  {3}{12}{13}   {2}{3}{12}{123}
  {3}{12}{23}   {2}{3}{13}{123}
  {1}{12}{123}  {1}{12}{23}{123}
  {1}{13}{123}  {1}{13}{23}{123}
  {2}{12}{123}  {2}{12}{13}{123}
  {2}{23}{123}  {2}{13}{23}{123}
  {3}{13}{123}  {3}{12}{13}{123}
  {3}{23}{123}  {3}{12}{23}{123}
		

Crossrefs

The non-covering version is A330282.

Costrict (or T_0) covering set-systems are A059201.

BII-numbers of fully chiral set-systems are A330226.

Non-isomorphic, fully chiral multiset partitions are A330227.

Fully chiral partitions are counted by A330228.

Fully chiral covering set-systems are A330229.

Fully chiral factorizations are A330235.

MM-numbers of fully chiral multisets of multisets are A330236.

Cf. A000612, A055621, A083323, A283877, A319559, A319564, A330224, A330234.

Programs

Mathematica

graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&Length[graprms[#]]==n!&]],{n,0,3}]

Formula

Binomial transform is A330282.