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A327358
Triangle read by rows where T(n,k) is the number of unlabeled antichains of nonempty sets covering n vertices with vertex-connectivity >= k.
6
1, 1, 0, 2, 1, 0, 5, 3, 2, 0, 20, 14, 10, 6, 0, 180, 157, 128, 91, 54, 0
OFFSET
0,4
COMMENTS
An antichain is a set of sets, none of which is a subset of any other. It is covering if there are no isolated vertices.
The vertex-connectivity of a set-system is the minimum number of vertices that must be removed (along with any empty or duplicate edges) to obtain a non-connected set-system or singleton. Note that this means a single node has vertex-connectivity 0.
If empty edges are allowed, we have T(0,0) = 2.
EXAMPLE
Triangle begins:
1
1 0
2 1 0
5 3 2 0
20 14 10 6 0
180 157 128 91 54 0
Non-isomorphic representatives of the antichains counted in row n = 4:
{1234} {1234} {1234} {1234}
{1}{234} {12}{134} {123}{124} {12}{134}{234}
{12}{34} {123}{124} {12}{13}{234} {123}{124}{134}
{12}{134} {12}{13}{14} {12}{134}{234} {12}{13}{14}{234}
{123}{124} {12}{13}{24} {123}{124}{134} {123}{124}{134}{234}
{1}{2}{34} {12}{13}{234} {12}{13}{24}{34} {12}{13}{14}{23}{24}{34}
{2}{13}{14} {12}{134}{234} {12}{13}{14}{234}
{12}{13}{14} {123}{124}{134} {12}{13}{14}{23}{24}
{12}{13}{24} {12}{13}{14}{23} {123}{124}{134}{234}
{1}{2}{3}{4} {12}{13}{24}{34} {12}{13}{14}{23}{24}{34}
{12}{13}{234} {12}{13}{14}{234}
{12}{134}{234} {12}{13}{14}{23}{24}
{123}{124}{134} {123}{124}{134}{234}
{4}{12}{13}{23} {12}{13}{14}{23}{24}{34}
{12}{13}{14}{23}
{12}{13}{24}{34}
{12}{13}{14}{234}
{12}{13}{14}{23}{24}
{123}{124}{134}{234}
{12}{13}{14}{23}{24}{34}
CROSSREFS
Column k = 0 is A261005, or A006602 if empty edges are allowed.
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.
Column k = n - 1 is A327425 (cointersecting).
The labeled version is A327350.
Negated first differences of rows are A327359.
Sequence in context: A073583 A324162 A060136 * A256664 A226783 A245972
KEYWORD
nonn,tabl,more
AUTHOR
Gus Wiseman, Sep 09 2019
STATUS
approved