A261005 -id:A261005 - 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

A326874 BII-numbers of abstract simplicial complexes.

Original entry on oeis.org

0, 1, 2, 3, 7, 8, 9, 10, 11, 15, 25, 27, 31, 42, 43, 47, 59, 63, 127, 128, 129, 130, 131, 135, 136, 137, 138, 139, 143, 153, 155, 159, 170, 171, 175, 187, 191, 255, 385, 387, 391, 393, 395, 399, 409, 411, 415, 427, 431, 443, 447, 511, 642, 643, 647, 650, 651, 655

Offset: 1

Gus Wiseman, Jul 29 2019

nonn

An abstract simplicial complex is a set of finite nonempty sets (edges) that is closed under taking a nonempty subset of any edge.
A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.
The enumeration of abstract simplicial complexes by number of covered vertices is given by A307249.

			The sequence of all abstract simplicial complexes together with their BII-numbers begins:
    0: {}
    1: {{1}}
    2: {{2}}
    3: {{1},{2}}
    7: {{1},{2},{1,2}}
    8: {{3}}
    9: {{1},{3}}
   10: {{2},{3}}
   11: {{1},{2},{3}}
   15: {{1},{2},{1,2},{3}}
   25: {{1},{3},{1,3}}
   27: {{1},{2},{3},{1,3}}
   31: {{1},{2},{3},{1,2},{1,3}}
   42: {{2},{3},{2,3}}
   43: {{1},{2},{3},{2,3}}
   47: {{1},{2},{3},{1,2},{2,3}}
   59: {{1},{2},{3},{1,3},{2,3}}
   63: {{1},{2},{3},{1,2},{1,3},{2,3}}
  127: {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
  128: {{4}}
  129: {{1},{4}}

Wikipedia, Abstract simplicial complex

Cf. A006126, A014466, A029931, A048793, A102896, A261005, A307249, A326031, A326872, A326876.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[0,100],SubsetQ[bpe/@bpe[#],DeleteCases[Union@@Subsets/@bpe/@bpe[#],{}]]&]

A327425 Number of unlabeled antichains of nonempty sets covering n vertices where every two vertices appear together in some edge (cointersecting).

Original entry on oeis.org

1, 1, 1, 2, 6, 54

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.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 6 antichains:
    {1}  {12}  {123}         {1234}
               {12}{13}{23}  {12}{134}{234}
                             {124}{134}{234}
                             {12}{13}{14}{234}
                             {123}{124}{134}{234}
                             {12}{13}{14}{23}{24}{34}

The labeled version is A327020.
Unlabeled covering antichains are A261005.
The weighted version is A327060.
Cf. A006126, A014466, A055621, A293606, A293993, A305844, A307249, A319639, A326704, A327057, A327058, A327358, A327359.

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).

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

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A326874 BII-numbers of abstract simplicial complexes.

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A327425 Number of unlabeled antichains of nonempty sets covering n vertices where every two vertices appear together in some edge (cointersecting).

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

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