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%I #7 Sep 10 2019 19:58:02
%S 1,1,0,2,1,0,5,3,2,0,20,14,10,6,0,180,157,128,91,54,0
%N Triangle read by rows where T(n,k) is the number of unlabeled antichains of nonempty sets covering n vertices with vertex-connectivity >= k.
%C 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.
%C 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.
%C If empty edges are allowed, we have T(0,0) = 2.
%e Triangle begins:
%e 1
%e 1 0
%e 2 1 0
%e 5 3 2 0
%e 20 14 10 6 0
%e 180 157 128 91 54 0
%e Non-isomorphic representatives of the antichains counted in row n = 4:
%e {1234} {1234} {1234} {1234}
%e {1}{234} {12}{134} {123}{124} {12}{134}{234}
%e {12}{34} {123}{124} {12}{13}{234} {123}{124}{134}
%e {12}{134} {12}{13}{14} {12}{134}{234} {12}{13}{14}{234}
%e {123}{124} {12}{13}{24} {123}{124}{134} {123}{124}{134}{234}
%e {1}{2}{34} {12}{13}{234} {12}{13}{24}{34} {12}{13}{14}{23}{24}{34}
%e {2}{13}{14} {12}{134}{234} {12}{13}{14}{234}
%e {12}{13}{14} {123}{124}{134} {12}{13}{14}{23}{24}
%e {12}{13}{24} {12}{13}{14}{23} {123}{124}{134}{234}
%e {1}{2}{3}{4} {12}{13}{24}{34} {12}{13}{14}{23}{24}{34}
%e {12}{13}{234} {12}{13}{14}{234}
%e {12}{134}{234} {12}{13}{14}{23}{24}
%e {123}{124}{134} {123}{124}{134}{234}
%e {4}{12}{13}{23} {12}{13}{14}{23}{24}{34}
%e {12}{13}{14}{23}
%e {12}{13}{24}{34}
%e {12}{13}{14}{234}
%e {12}{13}{14}{23}{24}
%e {123}{124}{134}{234}
%e {12}{13}{14}{23}{24}{34}
%Y Column k = 0 is A261005, or A006602 if empty edges are allowed.
%Y Column k = 1 is A261006 (clutters), if we assume A261006(0) = A261006(1) = 0.
%Y Column k = 2 is A305028 (blobs), if we assume A305028(0) = A305028(2) = 0.
%Y Column k = n - 1 is A327425 (cointersecting).
%Y The labeled version is A327350.
%Y Negated first differences of rows are A327359.
%Y Cf. A006126, A055621, A120338, A293606, A293993, A327334, A327351, A327356.
%K nonn,tabl,more
%O 0,4
%A _Gus Wiseman_, Sep 09 2019
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