A327425 -id:A327425 - cp's OEIS frontend

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.

1, 1, 0, 2, 1, 0, 5, 3, 2, 0, 20, 14, 10, 6, 0, 180, 157, 128, 91, 54, 0

Offset: 0

Gus Wiseman, Sep 09 2019

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.

			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}

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.
Cf. A006126, A055621, A120338, A293606, A293993, A327334, A327351, A327356.

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.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 2, 1, 2, 0, 6, 4, 4, 6, 0, 23, 29, 37, 37, 54, 0

Offset: 0

Gus Wiseman, Sep 10 2019

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.

			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}

Row sums are A261005, or A006602 if empty edges are allowed.
Column k = 0 is A327426.
Column k = 1 is A327436.
Column k = n - 1 is A327425.
The labeled version is A327351.
Cf. A003465, A006126, A014466, A048143, A293993, A323818, A326704, A327125, A327334, A327336, A327350, A327358.

cp's OEIS Frontend

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.

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