A305857 -id:A305857 - cp's OEIS frontend

A329628 Smallest BII-number of an intersecting antichain with n edges.

0, 1, 20, 52, 2880, 275520

Offset: 0

Gus Wiseman, Nov 28 2019

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 set-system (finite set of finite nonempty sets of positive integers) 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. Elements of a set-system are sometimes called edges.

A set-system is intersecting if no two edges are disjoint. It is an antichain if no edge is a proper subset of any other.

			The sequence of terms together with their corresponding set-systems begins:
       0: {}
       1: {{1}}
      20: {{1,2},{1,3}}
      52: {{1,2},{1,3},{2,3}}
    2880: {{1,2,3},{1,4},{2,4},{3,4}}
  275520: {{1,2,3},{1,2,4},{1,3,4},{2,3,4},{1,2,5}}

The not necessarily intersecting version is A329626.
MM-numbers of intersecting antichains are A329366.
BII-numbers of antichains are A326704.
BII-numbers of intersecting set-systems are A326910.
BII-numbers of intersecting antichains are A329561.
Covering intersecting antichains of sets are A305844.
Non-isomorphic intersecting antichains of multisets are A306007.
Cf. A000120, A048793, A070939, A072639, A316476, A305857, A326031, A326361, A326912, A329560.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
First/@GatherBy[Select[Range[0,10000],stableQ[bpe/@bpe[#],SubsetQ[#1,#2]||Intersection[#1,#2]=={}&]&],Length[bpe[#]]&]

A305855 Number of unlabeled spanning intersecting antichains on n vertices.

Original entry on oeis.org

1, 1, 1, 3, 9, 72, 3441, 47170585

Offset: 0

Gus Wiseman, Jun 11 2018

An intersecting antichain S is a finite set of finite nonempty sets (edges), any two of which have a nonempty intersection, and none of which is a subset of any other. S is spanning if every vertex is contained in some edge.

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

Cf. A001206, A006126, A051185, A261006, A283877, A304998, A305843, A305844, A305854-A305857.

a(n) = A305857(n) - A305857(n-1) for n > 0. - Andrew Howroyd, Aug 13 2019

a(6) from Andrew Howroyd, Aug 13 2019

a(7) from Brendan McKay, May 11 2020

A305856 Number of unlabeled intersecting set-systems on up to n vertices.

Original entry on oeis.org

1, 2, 4, 14, 124, 14992, 1289845584

Offset: 0

Gus Wiseman, Jun 11 2018

An intersecting set-system is a finite set of finite nonempty sets (edges), any two of which have a nonempty intersection.

			Non-isomorphic representatives of the a(3) = 14 intersecting set-systems:
  {}
  {{1}}
  {{1,2}}
  {{1,2,3}}
  {{2},{1,2}}
  {{3},{1,2,3}}
  {{1,3},{2,3}}
  {{2,3},{1,2,3}}
  {{3},{1,3},{2,3}}
  {{3},{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3}}
  {{1,3},{2,3},{1,2,3}}
  {{3},{1,3},{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3},{1,2,3}}

Cf. A001206, A006126, A051185, A261006, A283877, A304982, A304996, A304998, A305843, A305844, A305854-A305857.

a(5) from Andrew Howroyd, Aug 12 2019

a(6) from Bert Dobbelaere, Apr 28 2025

A305935 Number of labeled spanning intersecting set-systems on n vertices with no singletons.

Original entry on oeis.org

1, 0, 1, 12, 809, 1146800, 899927167353, 291136684655893185321964, 14704020783497694096988185391720223222562121969, 12553242487939982849962414795232892198542733492886483991398790450208264017757788101836749760

Offset: 0

Gus Wiseman, Jun 15 2018

nonn

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

			The a(3) = 12 spanning intersecting set-systems with no singletons:
{{1,2,3}}
{{1,2},{1,3}}
{{1,2},{2,3}}
{{1,3},{2,3}}
{{1,2},{1,2,3}}
{{1,3},{1,2,3}}
{{2,3},{1,2,3}}
{{1,2},{1,3},{2,3}}
{{1,2},{1,3},{1,2,3}}
{{1,2},{2,3},{1,2,3}}
{{1,3},{2,3},{1,2,3}}
{{1,2},{1,3},{2,3},{1,2,3}}

Cf. A001206, A006126, A051185, A048143, A058891, A305001, A305843, A305844, A305854-A305857, A305999, A306000, A306001, A306008.

a(n) = A305843(n) - n * A003465(n-1).

Inverse binomial transform of A306000. - Andrew Howroyd, Aug 12 2019

a(6)-a(8) from Giovanni Resta, Jun 20 2018

a(9) from Andrew Howroyd, Aug 12 2019

A306000 Number of labeled intersecting set-systems with no singletons covering some subset of {1,...,n}.

Original entry on oeis.org

1, 1, 2, 16, 864, 1150976, 899934060544, 291136684662192699604992, 14704020783497694096990514485197495566069661696, 12553242487939982849962414795232892198542733625222671042878037323112413463887484853594095616

Offset: 0

Gus Wiseman, Jun 16 2018

nonn

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.

			The a(3) = 16 set-systems:
  {}
  {{1,2}}
  {{1,3}}
  {{2,3}}
  {{1,2,3}}
  {{1,2},{1,3}}
  {{1,2},{2,3}}
  {{1,3},{2,3}}
  {{1,2},{1,2,3}}
  {{1,3},{1,2,3}}
  {{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,3},{1,2,3}}
  {{1,2},{2,3},{1,2,3}}
  {{1,3},{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3},{1,2,3}}

Cf. A001206, A006126, A051185, A048143, A058891, A305001, A305843, A305844, A305854-A305857, A305935, A305999, A306001.

a(n) = A051185(n) - n*2^(2^(n-1)-1). - Andrew Howroyd, Aug 12 2019

a(6)-a(9) from Andrew Howroyd, Aug 12 2019

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

Original entry on oeis.org

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

cp's OEIS Frontend

A329628 Smallest BII-number of an intersecting antichain with n edges.

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A305855 Number of unlabeled spanning intersecting antichains on n vertices.

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A305856 Number of unlabeled intersecting set-systems on up to n vertices.

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A305935 Number of labeled spanning intersecting set-systems on n vertices with no singletons.

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A306000 Number of labeled intersecting set-systems with no singletons covering some subset of {1,...,n}.

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

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