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

1, 1, 2, 42, 21336

Offset

0,3

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.

Links

Table of n, a(n) for n=0..4.

Formula

Binomial transform is A330282.

Example

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}

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}]

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.

Sequence in context: A193272 A193273 A182192 * A039622 A130506 A273399

Adjacent sequences: A330226 A330227 A330228 * A330230 A330231 A330232

Keyword

nonn,more

Author

Gus Wiseman, Dec 08 2019

Status

approved