A326948 Number of connected T_0 set-systems on n vertices.

1, 1, 3, 86, 31302, 2146841520, 9223371978880250448, 170141183460469231408869283342774399392, 57896044618658097711785492504343953919148780260559635830120038252613826101856

Offset

0,3

Comments

The dual of a set-system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The T_0 condition means that the dual is strict (no repeated edges).

Links

Andrew Howroyd, Table of n, a(n) for n = 0..11

Formula

Logarithmic transform of A059201.

Example

The a(3) = 86 set-systems:
  {12}{13}         {1}{2}{13}{123}     {1}{2}{3}{13}{23}
  {12}{23}         {1}{2}{23}{123}     {1}{2}{3}{13}{123}
  {13}{23}         {1}{3}{12}{13}      {1}{2}{3}{23}{123}
  {1}{2}{123}      {1}{3}{12}{23}      {1}{2}{12}{13}{23}
  {1}{3}{123}      {1}{3}{12}{123}     {1}{2}{12}{13}{123}
  {1}{12}{13}      {1}{3}{13}{23}      {1}{2}{12}{23}{123}
  {1}{12}{23}      {1}{3}{13}{123}     {1}{2}{13}{23}{123}
  {1}{12}{123}     {1}{3}{23}{123}     {1}{3}{12}{13}{23}
  {1}{13}{23}      {1}{12}{13}{23}     {1}{3}{12}{13}{123}
  {1}{13}{123}     {1}{12}{13}{123}    {1}{3}{12}{23}{123}
  {2}{3}{123}      {1}{12}{23}{123}    {1}{3}{13}{23}{123}
  {2}{12}{13}      {1}{13}{23}{123}    {1}{12}{13}{23}{123}
  {2}{12}{23}      {2}{3}{12}{13}      {2}{3}{12}{13}{23}
  {2}{12}{123}     {2}{3}{12}{23}      {2}{3}{12}{13}{123}
  {2}{13}{23}      {2}{3}{12}{123}     {2}{3}{12}{23}{123}
  {2}{23}{123}     {2}{3}{13}{23}      {2}{3}{13}{23}{123}
  {3}{12}{13}      {2}{3}{13}{123}     {2}{12}{13}{23}{123}
  {3}{12}{23}      {2}{3}{23}{123}     {3}{12}{13}{23}{123}
  {3}{13}{23}      {2}{12}{13}{23}     {1}{2}{3}{12}{13}{23}
  {3}{13}{123}     {2}{12}{13}{123}    {1}{2}{3}{12}{13}{123}
  {3}{23}{123}     {2}{12}{23}{123}    {1}{2}{3}{12}{23}{123}
  {12}{13}{23}     {2}{13}{23}{123}    {1}{2}{3}{13}{23}{123}
  {12}{13}{123}    {3}{12}{13}{23}     {1}{2}{12}{13}{23}{123}
  {12}{23}{123}    {3}{12}{13}{123}    {1}{3}{12}{13}{23}{123}
  {13}{23}{123}    {3}{12}{23}{123}    {2}{3}{12}{13}{23}{123}
  {1}{2}{3}{123}   {3}{13}{23}{123}    {1}{2}{3}{12}{13}{23}{123}
  {1}{2}{12}{13}   {12}{13}{23}{123}
  {1}{2}{12}{23}   {1}{2}{3}{12}{13}
  {1}{2}{12}{123}  {1}{2}{3}{12}{23}
  {1}{2}{13}{23}   {1}{2}{3}{12}{123}

Mathematica

dual[eds_]:=Table[First/@Position[eds, x], {x, Union@@eds}];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[OrderedQ[#], UnsameQ@@#, Length[Intersection@@s[[#]]]>0]&]}, If[c=={}, s, csm[Sort[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n], {1, n}]], Union@@#==Range[n]&&Length[csm[#]]<=1&&UnsameQ@@dual[#]&]], {n, 0, 3}]

Crossrefs

The same with covering instead of connected is A059201, with unlabeled version A319637.

The non-T_0 version is A323818 (covering) or A326951 (not-covering).

The non-connected version is A326940, with unlabeled version A326946.

Cf. A000371, A003465, A245567, A316978, A319559, A319564, A326939, A326941, A326947.

Sequence in context: A185142 A279020 A302947 * A159053 A172494 A279131

Adjacent sequences: A326945 A326946 A326947 * A326949 A326950 A326951

Keyword

nonn

Author

Gus Wiseman, Aug 08 2019

Status

approved