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A327359 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.

%I A327359 #6 Sep 10 2019 19:58:09
%S A327359 1,1,0,1,1,0,2,1,2,0,6,4,4,6,0,23,29,37,37,54,0
%N A327359 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.
%C A327359 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 A327359 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 A327359 If empty edges are allowed, we have T(0,0) = 2.
%e A327359 Triangle begins:
%e A327359    1
%e A327359    1  0
%e A327359    1  1  0
%e A327359    2  1  2  0
%e A327359    6  4  4  6  0
%e A327359   23 29 37 37 54  0
%e A327359 Row n = 4 counts the following antichains:
%e A327359 {1}{234}      {14}{234}        {134}{234}           {1234}
%e A327359 {12}{34}      {13}{24}{34}     {13}{14}{234}        {12}{134}{234}
%e A327359 {1}{2}{34}    {14}{24}{34}     {12}{13}{24}{34}     {124}{134}{234}
%e A327359 {1}{24}{34}   {14}{23}{24}{34} {13}{14}{23}{24}{34} {12}{13}{14}{234}
%e A327359 {1}{2}{3}{4}                                        {123}{124}{134}{234}
%e A327359 {1}{23}{24}{34}                                     {12}{13}{14}{23}{24}{34}
%Y A327359 Row sums are A261005, or A006602 if empty edges are allowed.
%Y A327359 Column k = 0 is A327426.
%Y A327359 Column k = 1 is A327436.
%Y A327359 Column k = n - 1 is A327425.
%Y A327359 The labeled version is A327351.
%Y A327359 Cf. A003465, A006126, A014466, A048143, A293993, A323818, A326704, A327125, A327334, A327336, A327350, A327358.
%K A327359 nonn,tabl,more
%O A327359 0,7
%A A327359 _Gus Wiseman_, Sep 10 2019