A261006 -id:A261006 - cp's OEIS frontend

A306001 Number of unlabeled intersecting set-systems with no singletons on up to n vertices.

1, 1, 2, 8, 84, 13000

Offset: 0

Gus Wiseman, Jun 16 2018

An intersecting set-system S is a finite set of finite nonempty sets (edges), any two of which have a nonempty intersection. A singleton is an edge containing only one vertex.

			Non-isomorphic representatives of the a(3) = 8 set-systems:
{}
{{1,2}}
{{1,2,3}}
{{1,3},{2,3}}
{{2,3},{1,2,3}}
{{1,2},{1,3},{2,3}}
{{1,3},{2,3},{1,2,3}}
{{1,2},{1,3},{2,3},{1,2,3}}

Cf. A001206, A051185, A048143, A261006, A058891, A261005, A304998, A305854-A305857, A305935, A305999, A306000.

a(n) = A305856(n) - A000612(n). - Andrew Howroyd, Aug 12 2019

a(5) from Andrew Howroyd, Aug 12 2019

A309667 Number of non-isomorphic connected set-systems on up to n vertices.

Original entry on oeis.org

1, 2, 5, 35, 1947, 18664537, 12813206150464222, 33758171486592987151274638818642016, 1435913805026242504952006868879460423801146743462225386062178112354069599

Offset: 0

Gus Wiseman, Aug 11 2019

nonn

A set-system is a finite set of finite nonempty sets.

			Non-isomorphic representatives of the a(0) = 1 through a(2) = 5 set-systems:
  {}  {}     {}
      {{1}}  {{1}}
             {{1,2}}
             {{2},{1,2}}
             {{1},{2},{1,2}}

Alois P. Heinz, Table of n, a(n) for n = 0..12

The covering case is A323819 (first differences).
The BII-numbers of connected set-systems are A326749.
The labeled version is A326964.
Cf. A000371, A000612, A001187, A007718, A058891, A092918, A261006, A300913, A323818.

A327436 Number of connected, unlabeled antichains of nonempty subsets of {1..n} covering n vertices with at least one cut-vertex (vertex-connectivity 1).

Original entry on oeis.org

0, 0, 1, 1, 4, 29

Offset: 0

Gus Wiseman, Sep 11 2019

nonn

			Non-isomorphic representatives of the a(2) = 1 through a(5) = 29 antichains:
  {12}  {12}{13}  {12}{134}         {12}{1345}
                  {12}{13}{14}      {123}{145}
                  {12}{13}{24}      {12}{13}{145}
                  {12}{13}{14}{23}  {12}{13}{245}
                                    {13}{24}{125}
                                    {13}{124}{125}
                                    {14}{123}{235}
                                    {12}{13}{14}{15}
                                    {12}{13}{14}{25}
                                    {12}{13}{24}{35}
                                    {12}{13}{14}{235}
                                    {12}{13}{23}{145}
                                    {12}{13}{45}{234}
                                    {12}{14}{23}{135}
                                    {12}{15}{134}{234}
                                    {15}{23}{124}{134}
                                    {15}{123}{124}{134}
                                    {15}{123}{124}{234}
                                    {12}{13}{14}{15}{23}
                                    {12}{13}{14}{23}{25}
                                    {12}{13}{14}{23}{45}
                                    {12}{13}{15}{24}{34}
                                    {12}{13}{14}{15}{234}
                                    {12}{13}{14}{25}{234}
                                    {12}{13}{14}{15}{23}{24}
                                    {12}{13}{14}{15}{23}{45}
                                    {12}{13}{14}{23}{24}{35}
                                    {15}{123}{124}{134}{234}
                                    {12}{13}{14}{15}{23}{24}{34}

Column k = 1 of A327359.
The labeled version is A327356.
Cf. A006602, A014466, A048143, A261005, A326704, A326786, A327112, A327114, A327426, A327334, A327336, A327350, A327351, A327358.

a(n > 2) = A261006(n) - A305028(n).

A305005 Number of labeled clutters (connected antichains) spanning some subset of {1,...,n} without singleton edges.

Original entry on oeis.org

1, 1, 2, 9, 111, 6829, 7783192, 2414627236071, 56130437209370100252463

Offset: 0

Gus Wiseman, May 23 2018

nonn

			The a(3) = 9 clutters:
  {}
  {{1,2}}
  {{1,3}}
  {{2,3}}
  {{1,2,3}}
  {{1,2},{1,3}}
  {{1,2},{2,3}}
  {{1,3},{2,3}}
  {{1,2},{1,3},{2,3}}

Gus Wiseman, Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons.

Cf. A048143, A198085, A261006, A304912, A304981, A304982, A304983, A304984, A304985, A304986.

Binomial transform of A048143 if we assume A048143(1) = 0.

a(n) = A198085(n) - n + 1. - Gus Wiseman, Jun 11 2018

A327807 Triangle read by rows where T(n,k) is the number of unlabeled antichains of sets with n vertices and vertex-connectivity >= k.

Original entry on oeis.org

1, 2, 0, 4, 1, 0, 9, 3, 2, 0, 29, 14, 10, 6, 0, 209, 157, 128, 91, 54, 0

Offset: 0

Gus Wiseman, Sep 26 2019

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

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.

			Triangle begins:
    1
    2   0
    4   1   0
    9   3   2   0
   29  14  10   6   0
  209 157 128  91  54   0

Column k = 0 is A306505.
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.
Except for the first column, same as A327358 (the covering case).
The labeled version is A327806.
Cf. A014466, A293606, A326704, A326950, A327062, A327334, A327350, A327351, A327805, A327808.

cp's OEIS Frontend

A306001 Number of unlabeled intersecting set-systems with no singletons on up to n vertices.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A309667 Number of non-isomorphic connected set-systems on up to n vertices.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A327436 Number of connected, unlabeled antichains of nonempty subsets of {1..n} covering n vertices with at least one cut-vertex (vertex-connectivity 1).

Views

Author

Keywords

Examples

Crossrefs

Formula

A305005 Number of labeled clutters (connected antichains) spanning some subset of {1,...,n} without singleton edges.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A327807 Triangle read by rows where T(n,k) is the number of unlabeled antichains of sets with n vertices and vertex-connectivity >= k.

Views

Author

Keywords

Comments

Examples

Crossrefs