A326868 Number of connected connectedness systems on n vertices.

1, 1, 4, 64, 6048, 8064000, 1196002238976

Offset

0,3

Comments

We define a connectedness system (investigated by Vim van Dam in 2002) to be a set of finite nonempty sets (edges) that is closed under taking the union of any two overlapping edges. It is connected if it is empty or contains an edge with all the vertices.

Links

Table of n, a(n) for n=0..6.

Gus Wiseman, Every Clutter Is a Tree of Blobs, The Mathematica Journal, Vol. 19, 2017.

Formula

a(n > 1) = 2^n * A072447(n).

Logarithmic transform of A326870.

Example

The a(3) = 64 connected connectedness systems:
  {{123}}              {{1}{123}}
  {{12}{123}}          {{2}{123}}
  {{13}{123}}          {{3}{123}}
  {{23}{123}}          {{1}{12}{123}}
  {{12}{13}{123}}      {{1}{13}{123}}
  {{12}{23}{123}}      {{1}{23}{123}}
  {{13}{23}{123}}      {{2}{12}{123}}
  {{12}{13}{23}{123}}  {{2}{13}{123}}
                       {{2}{23}{123}}
                       {{3}{12}{123}}
                       {{3}{13}{123}}
                       {{3}{23}{123}}
                       {{1}{12}{13}{123}}
                       {{1}{12}{23}{123}}
                       {{1}{13}{23}{123}}
                       {{2}{12}{13}{123}}
                       {{2}{12}{23}{123}}
                       {{2}{13}{23}{123}}
                       {{3}{12}{13}{123}}
                       {{3}{12}{23}{123}}
                       {{3}{13}{23}{123}}
                       {{1}{12}{13}{23}{123}}
                       {{2}{12}{13}{23}{123}}
                       {{3}{12}{13}{23}{123}}
.
  {{1}{2}{123}}              {{1}{2}{3}{123}}
  {{1}{3}{123}}              {{1}{2}{3}{12}{123}}
  {{2}{3}{123}}              {{1}{2}{3}{13}{123}}
  {{1}{2}{12}{123}}          {{1}{2}{3}{23}{123}}
  {{1}{2}{13}{123}}          {{1}{2}{3}{12}{13}{123}}
  {{1}{2}{23}{123}}          {{1}{2}{3}{12}{23}{123}}
  {{1}{3}{12}{123}}          {{1}{2}{3}{13}{23}{123}}
  {{1}{3}{13}{123}}          {{1}{2}{3}{12}{13}{23}{123}}
  {{1}{3}{23}{123}}
  {{2}{3}{12}{123}}
  {{2}{3}{13}{123}}
  {{2}{3}{23}{123}}
  {{1}{2}{12}{13}{123}}
  {{1}{2}{12}{23}{123}}
  {{1}{2}{13}{23}{123}}
  {{1}{3}{12}{13}{123}}
  {{1}{3}{12}{23}{123}}
  {{1}{3}{13}{23}{123}}
  {{2}{3}{12}{13}{123}}
  {{2}{3}{12}{23}{123}}
  {{2}{3}{13}{23}{123}}
  {{1}{2}{12}{13}{23}{123}}
  {{1}{3}{12}{13}{23}{123}}
  {{2}{3}{12}{13}{23}{123}}

Mathematica

Table[Length[Select[Subsets[Subsets[Range[n], {1, n}]], n==0||MemberQ[#, Range[n]]&&SubsetQ[#, Union@@@Select[Tuples[#, 2], Intersection@@#!={}&]]&]], {n, 0, 4}]

Crossrefs

The case without singletons is A072447.

The not necessarily connected case is A326866.

The unlabeled case is A326869.

The BII-numbers of these set-systems are A326879.

Cf. A072445, A072446, A102896, A306445, A323818, A326867, A326870, A326872.

Sequence in context: A348315 A053923 A359231 * A211214 A229867 A362383

Adjacent sequences: A326865 A326866 A326867 * A326869 A326870 A326871

Keyword

nonn,more

Author

Gus Wiseman, Jul 29 2019

Extensions

a(6) corrected by Christian Sievers, Oct 28 2023

Status

approved