A326961 Number of set-systems covering n vertices where every vertex is the unique common element of some subset of the edges, also called covering T_1 set-systems.

1, 1, 2, 36, 19020, 2010231696, 9219217412568364176, 170141181796805105960861096082778425120, 57896044618658097536026644159052312977171804852352892309392604715987334365792

Offset: 0

Author

Gus Wiseman, Aug 12 2019

Keywords

• nonn

Comments

Same as A059523 except with a(1) = 1 instead of 2.

Alternatively, these are set-systems covering n vertices whose dual is a (strict) antichain. A set-system is a finite set of finite nonempty sets. The dual of a set-system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. An antichain is a set of sets, none of which is a subset of any other.

Examples

			The a(3) = 36 set-systems:
  {{1}{2}{3}}        {{12}{13}{23}{123}}     {{2}{3}{12}{13}{23}}
  {{12}{13}{23}}     {{1}{2}{3}{12}{13}}     {{2}{3}{12}{13}{123}}
  {{1}{2}{3}{12}}    {{1}{2}{3}{12}{23}}     {{2}{12}{13}{23}{123}}
  {{1}{2}{3}{13}}    {{1}{2}{3}{13}{23}}     {{3}{12}{13}{23}{123}}
  {{1}{2}{3}{23}}    {{1}{2}{12}{13}{23}}    {{1}{2}{3}{12}{13}{23}}
  {{1}{2}{13}{23}}   {{1}{2}{3}{12}{123}}    {{1}{2}{3}{12}{13}{123}}
  {{1}{2}{3}{123}}   {{1}{2}{3}{13}{123}}    {{1}{2}{3}{12}{23}{123}}
  {{1}{3}{12}{23}}   {{1}{2}{3}{23}{123}}    {{1}{2}{3}{13}{23}{123}}
  {{2}{3}{12}{13}}   {{1}{3}{12}{13}{23}}    {{1}{2}{12}{13}{23}{123}}
  {{1}{12}{13}{23}}  {{1}{2}{13}{23}{123}}   {{1}{3}{12}{13}{23}{123}}
  {{2}{12}{13}{23}}  {{1}{3}{12}{23}{123}}   {{2}{3}{12}{13}{23}{123}}
  {{3}{12}{13}{23}}  {{1}{12}{13}{23}{123}}  {{1}{2}{3}{12}{13}{23}{123}}
		

Crossrefs

Covering set-systems are A003465.

Covering T_0 set-systems are A059201.

The version with empty edges allowed is A326960.

The non-covering version is A326965.

Covering set-systems whose dual is a weak antichain are A326970.

The unlabeled version is A326974.

The BII-numbers of T_1 set-systems are A326979.

Cf. A058891, A059052, A059523, A323818, A326972, A326973, A326976, A326977.

Programs

Mathematica

tmQ[eds_]:=Union@@Select[Intersection@@@Rest[Subsets[eds]],Length[#]==1&]==Union@@eds;
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&tmQ[#]&]],{n,0,3}]

Formula

Inverse binomial transform of A326965.