A326970 Number of set-systems covering n vertices whose dual is a weak antichain.

1, 1, 3, 43, 19251

Offset

0,3

Comments

A set-system is a finite set of finite nonempty sets. The dual of a set-system has, for each vertex, one edges 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}}. A weak antichain is a multiset of sets, none of which is a proper subset of any other.

Links

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

Formula

Inverse binomial transform of A326968.

Example

The a(3) = 43 set-systems:
  {123}  {1}{23}  {1}{2}{3}     {1}{2}{3}{12}
         {2}{13}  {12}{13}{23}  {1}{2}{3}{13}
         {3}{12}  {1}{23}{123}  {1}{2}{3}{23}
                  {2}{13}{123}  {1}{2}{13}{23}
                  {3}{12}{123}  {1}{2}{3}{123}
                                {1}{3}{12}{23}
                                {2}{3}{12}{13}
                                {1}{12}{13}{23}
                                {2}{12}{13}{23}
                                {3}{12}{13}{23}
                                {12}{13}{23}{123}
.
  {1}{2}{3}{12}{13}     {1}{2}{3}{12}{13}{23}    {1}{2}{3}{12}{13}{23}{123}
  {1}{2}{3}{12}{23}     {1}{2}{3}{12}{13}{123}
  {1}{2}{3}{13}{23}     {1}{2}{3}{12}{23}{123}
  {1}{2}{12}{13}{23}    {1}{2}{3}{13}{23}{123}
  {1}{2}{3}{12}{123}    {1}{2}{12}{13}{23}{123}
  {1}{2}{3}{13}{123}    {1}{3}{12}{13}{23}{123}
  {1}{2}{3}{23}{123}    {2}{3}{12}{13}{23}{123}
  {1}{3}{12}{13}{23}
  {2}{3}{12}{13}{23}
  {1}{2}{13}{23}{123}
  {1}{3}{12}{23}{123}
  {2}{3}{12}{13}{123}
  {1}{12}{13}{23}{123}
  {2}{12}{13}{23}{123}
  {3}{12}{13}{23}{123}

Mathematica

dual[eds_]:=Table[First/@Position[eds, x], {x, Union@@eds}];
stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];
Table[Length[Select[Subsets[Subsets[Range[n], {1, n}]], Union@@#==Range[n]&&stableQ[dual[#], SubsetQ]&]], {n, 0, 3}]

Crossrefs

Covering set-systems are A003465.

Covering set-systems whose dual is strict are A059201.

The T_1 case is A326961.

The BII-numbers of these set-systems are A326966.

The non-covering case is A326968.

The unlabeled version is A326973.

Cf. A006126, A059052, A059523, A326950, A326960, A326965, A326969, A326971, A326974, A326975, A326978.

Sequence in context: A190637 A287959 A346200 * A076361 A130408 A349646

Adjacent sequences: A326967 A326968 A326969 * A326971 A326972 A326973

Keyword

nonn,more

Author

Gus Wiseman, Aug 10 2019

Status

approved