A327437 -id:A327437 - cp's OEIS frontend

A327355 Number of antichains of nonempty subsets of {1..n} that are either non-connected or non-covering (spanning edge-connectivity 0).

1, 1, 4, 14, 83, 1232, 84625, 109147467, 38634257989625

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 spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (without removing incident vertices) to obtain a set-system that is disconnected or covers fewer vertices.

			The a(1) = 1 through a(3) = 14 antichains:
  {}  {}         {}
      {{1}}      {{1}}
      {{2}}      {{2}}
      {{1},{2}}  {{3}}
                 {{1,2}}
                 {{1,3}}
                 {{2,3}}
                 {{1},{2}}
                 {{1},{3}}
                 {{2},{3}}
                 {{1},{2,3}}
                 {{2},{1,3}}
                 {{3},{1,2}}
                 {{1},{2},{3}}

Column k = 0 of A327352.
The covering case is A120338.
The unlabeled version is A327437.
The non-spanning edge-connectivity version is A327354.
Cf. A014466, A327071, A327148, A327353, A327426.

a(n) = A120338(n) + A014466(n) - A006126(n).

A327426 Number of non-connected, unlabeled, antichain covers of {1..n} (vertex-connectivity 0).

Original entry on oeis.org

1, 1, 1, 2, 6, 23, 201, 16345

Offset: 0

Gus Wiseman, Sep 11 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. A singleton is not considered connected.

The vertex-connectivity of a set-system is the minimum number of vertices that must be removed (along with any resulting empty edges) to obtain a non-connected set-system or singleton. Note that this means a single node has vertex-connectivity 0.

			Non-isomorphic representatives of the a(2) = 1 through a(5) = 23 antichains:
    {1}{2}  {1}{23}    {1}{234}         {1}{2345}
            {1}{2}{3}  {12}{34}         {12}{345}
                       {1}{2}{34}       {1}{2}{345}
                       {1}{24}{34}      {1}{23}{45}
                       {1}{2}{3}{4}     {12}{35}{45}
                       {1}{23}{24}{34}  {1}{25}{345}
                                        {1}{2}{3}{45}
                                        {1}{245}{345}
                                        {1}{2}{35}{45}
                                        {1}{2}{3}{4}{5}
                                        {1}{24}{35}{45}
                                        {1}{25}{35}{45}
                                        {12}{34}{35}{45}
                                        {1}{24}{25}{345}
                                        {1}{23}{245}{345}
                                        {1}{2}{34}{35}{45}
                                        {1}{235}{245}{345}
                                        {1}{23}{24}{35}{45}
                                        {1}{25}{34}{35}{45}
                                        {1}{23}{24}{25}{345}
                                        {1}{234}{235}{245}{345}
                                        {1}{24}{25}{34}{35}{45}
                                        {1}{23}{24}{25}{34}{35}{45}

Column k = 0 of A327359.
The labeled version is A120338.
The non-covering version is A327424 (partial sums).
Cf. A014466, A261005, A327354, A327355, A327437.

a(n > 1) = A261005(n) - A261006(n).

A327424 Number of unlabeled, non-connected or empty antichains of nonempty subsets of {1..n}.

Original entry on oeis.org

1, 1, 2, 4, 10, 33, 234, 16579

Offset: 0

Gus Wiseman, Sep 26 2019

An antichain is a set of nonempty sets, none of which is a subset of any other. A singleton is considered to be connected.

			Non-isomorphic representatives of the a(0) = 1 through a(4) = 10 antichains:
  {}  {}  {}         {}             {}
          {{1},{2}}  {{1},{2}}      {{1},{2}}
                     {{1},{2,3}}    {{1},{2,3}}
                     {{1},{2},{3}}  {{1},{2},{3}}
                                    {{1},{2,3,4}}
                                    {{1,2},{3,4}}
                                    {{1},{2},{3,4}}
                                    {{1},{2},{3},{4}}
                                    {{1},{2,4},{3,4}}
                                    {{1},{2,3},{2,4},{3,4}}

Partial sums of the positive-index terms of A327426.
The covering case is A327426.
The labeled version is A327354.
The labeled covering case is A120338.
Unlabeled antichains that are either not connected or not covering are A327437.
The case without empty antichains is A327808.
Cf. A014466, A293606, A326704, A327062, A327355.

A327438 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of unlabeled antichains of nonempty subsets of {1..n} with spanning edge-connectivity k.

Original entry on oeis.org

1, 1, 1, 3, 1, 6, 2, 1, 15, 7, 5, 2, 52, 53, 62, 31, 9, 1, 1

Offset: 0

Gus Wiseman, Sep 11 2019

An antichain is a set of sets, none of which is a subset of any other.

The spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (without removing incident vertices) to obtain a set-system that is disconnected or covers fewer vertices.

			Triangle begins:
   1
   1  1
   3  1
   6  2  1
  15  7  5  2
  52 53 62 31  9  1  1
The antichains counted in row n = 4 are the following:
  0             {1234}         {12}{134}{234}     {123}{124}{134}{234}
  {1}           {12}{134}      {123}{124}{134}    {12}{13}{14}{23}{24}{34}
  {12}          {123}{124}     {12}{13}{24}{34}
  {123}         {12}{13}{14}   {12}{13}{14}{234}
  {1}{2}        {12}{13}{24}   {12}{13}{14}{23}{24}
  {1}{23}       {12}{13}{234}
  {12}{13}      {12}{13}{14}{23}
  {1}{234}
  {12}{34}
  {1}{2}{3}
  {1}{2}{34}
  {2}{13}{14}
  {12}{13}{23}
  {1}{2}{3}{4}
  {4}{12}{13}{23}

Row sums are A306505.
Column k = 0 is A327437.
The labeled version is A327352.
Cf. A014466, A052446, A327062, A327071, A327103, A327111, A327144, A327352, A327353.

A327808 Number of unlabeled, disconnected, nonempty antichains of subsets of {1..n}.

Original entry on oeis.org

0, 0, 1, 3, 9, 32, 233, 16578

Offset: 0

Gus Wiseman, Sep 26 2019

An antichain is a set of nonempty sets, none of which is a subset of any other. A singleton is considered to be connected.

			Non-isomorphic representatives of the a(2) = 1 through a(4) = 9 antichains:
   {{1},{2}}  {{1},{2}}      {{1},{2}}
              {{1},{2,3}}    {{1},{2,3}}
              {{1},{2},{3}}  {{1},{2},{3}}
                             {{1},{2,3,4}}
                             {{1,2},{3,4}}
                             {{1},{2},{3,4}}
                             {{1},{2},{3},{4}}
                             {{2},{1,3},{1,4}}
                             {{4},{1,2},{1,3},{2,3}}

The labeled version is A327354 - 1.
The covering case is A327426.
Unlabeled antichains that are either not connected or not covering are A327437.
The version with empty antichains allowed is A327424.
Cf. A000372, A014466, A120338, A293606, A326704, A327062, A327355.

a(n) = A327424(n) - 1.

cp's OEIS Frontend

A327355 Number of antichains of nonempty subsets of {1..n} that are either non-connected or non-covering (spanning edge-connectivity 0).

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A327426 Number of non-connected, unlabeled, antichain covers of {1..n} (vertex-connectivity 0).

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A327424 Number of unlabeled, non-connected or empty antichains of nonempty subsets of {1..n}.

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A327438 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of unlabeled antichains of nonempty subsets of {1..n} with spanning edge-connectivity k.

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A327808 Number of unlabeled, disconnected, nonempty antichains of subsets of {1..n}.

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