A327358 - OEIS

%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