A327808 -id:A327808 - cp's OEIS frontend

A327424 Number of unlabeled, non-connected or empty antichains of nonempty subsets of {1..n}.

1, 1, 2, 4, 10, 33, 234, 16579

Offset: 0

Gus Wiseman, Sep 26 2019

An antichain is a set of nonempty sets, none of which is a subset of any other. A singleton is considered to be connected.

			Non-isomorphic representatives of the a(0) = 1 through a(4) = 10 antichains:
  {}  {}  {}         {}             {}
          {{1},{2}}  {{1},{2}}      {{1},{2}}
                     {{1},{2,3}}    {{1},{2,3}}
                     {{1},{2},{3}}  {{1},{2},{3}}
                                    {{1},{2,3,4}}
                                    {{1,2},{3,4}}
                                    {{1},{2},{3,4}}
                                    {{1},{2},{3},{4}}
                                    {{1},{2,4},{3,4}}
                                    {{1},{2,3},{2,4},{3,4}}

Partial sums of the positive-index terms of A327426.
The covering case is A327426.
The labeled version is A327354.
The labeled covering case is A120338.
Unlabeled antichains that are either not connected or not covering are A327437.
The case without empty antichains is A327808.
Cf. A014466, A293606, A326704, A327062, A327355.

A327807 Triangle read by rows where T(n,k) is the number of unlabeled antichains of sets with n vertices and vertex-connectivity >= k.

Original entry on oeis.org

1, 2, 0, 4, 1, 0, 9, 3, 2, 0, 29, 14, 10, 6, 0, 209, 157, 128, 91, 54, 0

Offset: 0

Gus Wiseman, Sep 26 2019

An antichain is a set of sets, none of which is a subset of any other.

The vertex-connectivity of a set-system is the minimum number of vertices that must be removed (along with any resulting empty edges) to obtain a non-connected set-system or singleton. Note that this means a single node has vertex-connectivity 0.

			Triangle begins:
    1
    2   0
    4   1   0
    9   3   2   0
   29  14  10   6   0
  209 157 128  91  54   0

Column k = 0 is A306505.
Column k = 1 is A261006 (clutters), if we assume A261006(0) = A261006(1) = 0.
Column k = 2 is A305028 (blobs), if we assume A305028(0) = A305028(2) = 0.
Except for the first column, same as A327358 (the covering case).
The labeled version is A327806.
Cf. A014466, A293606, A326704, A326950, A327062, A327334, A327350, A327351, A327805, A327808.

cp's OEIS Frontend

A327424 Number of unlabeled, non-connected or empty antichains of nonempty subsets of {1..n}.

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