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A327359
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Triangle read by rows where T(n,k) is the number of unlabeled antichains of nonempty sets covering n vertices with vertex-connectivity exactly k.
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5
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1, 1, 0, 1, 1, 0, 2, 1, 2, 0, 6, 4, 4, 6, 0, 23, 29, 37, 37, 54, 0
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OFFSET
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0,7
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COMMENTS
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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.
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LINKS
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EXAMPLE
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Triangle begins:
1
1 0
1 1 0
2 1 2 0
6 4 4 6 0
23 29 37 37 54 0
Row n = 4 counts the following antichains:
{1}{234} {14}{234} {134}{234} {1234}
{12}{34} {13}{24}{34} {13}{14}{234} {12}{134}{234}
{1}{2}{34} {14}{24}{34} {12}{13}{24}{34} {124}{134}{234}
{1}{24}{34} {14}{23}{24}{34} {13}{14}{23}{24}{34} {12}{13}{14}{234}
{1}{2}{3}{4} {123}{124}{134}{234}
{1}{23}{24}{34} {12}{13}{14}{23}{24}{34}
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CROSSREFS
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Cf. A003465, A006126, A014466, A048143, A293993, A323818, A326704, A327125, A327334, A327336, A327350, A327358.
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KEYWORD
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AUTHOR
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STATUS
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approved
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