A304985 -id:A304985 - cp's OEIS frontend

A305843 Number of labeled spanning intersecting set-systems on n vertices.

1, 1, 3, 27, 1245, 1308285, 912811093455, 291201248260060977862887, 14704022144627161780742038728709819246535634969, 12553242487940503914363982718112298267975272588471811456164576678961759219689708372356843289

Offset: 0

Gus Wiseman, Jun 11 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.

			The a(3) = 27 spanning intersecting set-systems:
{{1,2,3}}
{{1},{1,2,3}}
{{2},{1,2,3}}
{{3},{1,2,3}}
{{1,2},{1,3}}
{{1,2},{2,3}}
{{1,2},{1,2,3}}
{{1,3},{2,3}}
{{1,3},{1,2,3}}
{{2,3},{1,2,3}}
{{1},{1,2},{1,3}}
{{1},{1,2},{1,2,3}}
{{1},{1,3},{1,2,3}}
{{2},{1,2},{2,3}}
{{2},{1,2},{1,2,3}}
{{2},{2,3},{1,2,3}}
{{3},{1,3},{2,3}}
{{3},{1,3},{1,2,3}}
{{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},{1,2},{1,3},{1,2,3}}
{{2},{1,2},{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, A048143, A051185, A134958, A030019, A304985, A305844.

Length/@Table[Select[Subsets[Rest[Subsets[Range[n]]]],And[Union@@#==Range[n],FreeQ[Intersection@@@Tuples[#,2],{}]]&],{n,1,4}]

Inverse binomial transform of A051185.

A305844 Number of labeled spanning intersecting antichains on n vertices.

Original entry on oeis.org

1, 1, 1, 5, 50, 2330, 1407712, 229800077244, 423295097236295093695

Offset: 0

Gus Wiseman, Jun 11 2018

nonn

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.

			The a(3) = 5 spanning intersecting antichains:
{{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. A001206, A006126, A048143, A051185, A134958, A030019, A304985, A305843.

Length/@Table[Select[Subsets[Rest[Subsets[Range[n]]]],And[Union@@#==Range[n],FreeQ[Intersection@@@Tuples[#,2],{},{1}],Select[Tuples[#,2],UnsameQ@@#&&Complement@@#=={}&]=={}]&],{n,1,4}]

Inverse binomial transform of A001206(n + 1).

A304982 Number of unlabeled clutters (connected antichains) spanning up to n vertices with singleton edges allowed.

Original entry on oeis.org

1, 2, 5, 19, 137, 3053, 822526

Offset: 0

Gus Wiseman, May 23 2018

The initial terms 1, 2, 5, 19 are the same as A304981 but the remaining terms differ.

			Non-isomorphic representatives of the a(3) = 19 clutters:
{}
{{1}}
{{1,2}}
{{1,2,3}}
{{2},{1,2}}
{{1,3},{2,3}}
{{3},{1,2,3}}
{{1},{2},{1,2}}
{{1,2},{1,3},{2,3}}
{{3},{1,2},{2,3}}
{{3},{1,3},{2,3}}
{{2},{3},{1,2,3}}
{{1},{2},{3},{1,2,3}}
{{2},{3},{1,2},{1,3}}
{{3},{1,2},{1,3},{2,3}}
{{2},{3},{1,3},{2,3}}
{{1},{2},{3},{1,3},{2,3}}
{{2},{3},{1,2},{1,3},{2,3}}
{{1},{2},{3},{1,2},{1,3},{2,3}}

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

Partial sums of A304983.

a(5)-a(6) from Andrew Howroyd, Aug 14 2019

A304983 Number of unlabeled clutters (connected antichains) spanning n vertices with singleton edges allowed.

Original entry on oeis.org

1, 1, 3, 14, 118, 2916, 819473

Offset: 0

Gus Wiseman, May 23 2018

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

Cf. A048143, A198085, A261006, A304912, A304981, A304982, A304983, A304984, A304985, A304986, A304997.

Inverse Euler transform of A304997. - Andrew Howroyd, Aug 14 2019

a(5)-a(6) from Andrew Howroyd, Aug 14 2019

A304986 Number of labeled clutters (connected antichains) spanning some subset of {1,...,n}, if clutters of the form {{x}} are allowed for any vertex x.

Original entry on oeis.org

1, 2, 4, 12, 115, 6834, 7783198, 2414627236078, 56130437209370100252471

Offset: 0

Gus Wiseman, May 23 2018

nonn

			The a(3) = 12 clutters:
  {}
  {{1}}
  {{2}}
  {{3}}
  {{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}}

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

a(n > 0) = A198085(n) + 1.

a(n) = A305005(n) + n.

A304999 Number of labeled antichains of finite sets spanning n vertices with singleton edges allowed.

Original entry on oeis.org

1, 1, 5, 53, 1577, 212137, 496946349, 309068823607069, 14369391923126237496803793, 146629927766168786109802623629262590838145873

Offset: 0

Gus Wiseman, May 23 2018

Only the non-singleton edges are required to form an antichain.

			The a(2) = 5 antichains:
  {{1,2}}
  {{1},{2}}
  {{1},{1,2}}
  {{2},{1,2}}
  {{1},{2},{1,2}}

Aniruddha Biswas and Palash Sarkar, Counting unate and balanced monotone Boolean functions, arXiv:2304.14069 [math.CO], 2023.
Aniruddha Biswas and Palash Sarkar, Counting Unate and Monotone Boolean Functions Under Restrictions of Balancedness and Non-Degeneracy, J. Int. Seq. (2025) Vol. 28, Art. No. 25.3.4. See p. 4.
Gus Wiseman, Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons.

Cf. A006126, A014466, A261005, A304996, A304997, A304998, A304999, A305000, A305001.

Cf. A000372, A003182, A304985, A305844, A306505, A319721, A321679, A324167.

Exponential transform of A304985.

Inverse binomial transform of A305000. - Aniruddha Biswas, May 12 2024

a(5)-a(8) from Gus Wiseman, May 31 2018

a(9) from Aniruddha Biswas, May 12 2024

A304981 Number of unlabeled clutters (connected antichains) spanning up to n vertices without singleton edges.

Original entry on oeis.org

1, 1, 2, 5, 19, 176, 16118, 489996568

Offset: 0

Gus Wiseman, May 23 2018

nonn

			Non-isomorphic representatives of the a(3) = 5 clutters:
  {}
  {{1,2}}
  {{1,2,3}}
  {{1,3},{2,3}}
  {{1,2},{1,3},{2,3}}
Non-isomorphic representatives of the a(4) = 19 clutters:
  {}
  {{1,2}}
  {{1,2,3}}
  {{1,2,3,4}}
  {{1,3},{2,3}}
  {{1,4},{2,3,4}}
  {{1,3,4},{2,3,4}}
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,3,4},{2,3,4}}
  {{1,3},{1,4},{2,3,4}}
  {{1,3},{2,4},{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},{1,3},{2,4},{3,4}}
  {{1,4},{2,3},{2,4},{3,4}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
  {{1,3},{1,4},{2,3},{2,4},{3,4}}
  {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}

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

Partial sums of A261006(n > 0).

A304984 Number of labeled clutters (connected antichains) spanning some subset of {1,...,n} with singleton edges allowed.

Original entry on oeis.org

1, 2, 7, 56, 1533, 210302, 496838435, 309068803876372, 14369391923126181310256825

Offset: 0

Gus Wiseman, May 23 2018

nonn

			The a(2) = 7 clutters:
  {}
  {{1}}
  {{2}}
  {{1,2}}
  {{1},{1,2}}
  {{2},{1,2}}
  {{1},{2},{1,2}}

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

Binomial transform of A304985(n > 0).

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

cp's OEIS Frontend

A305843 Number of labeled spanning intersecting set-systems on n vertices.

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A305844 Number of labeled spanning intersecting antichains on n vertices.

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A304982 Number of unlabeled clutters (connected antichains) spanning up to n vertices with singleton edges allowed.

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A304983 Number of unlabeled clutters (connected antichains) spanning n vertices with singleton edges allowed.

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A304986 Number of labeled clutters (connected antichains) spanning some subset of {1,...,n}, if clutters of the form {{x}} are allowed for any vertex x.

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A304999 Number of labeled antichains of finite sets spanning n vertices with singleton edges allowed.

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A304981 Number of unlabeled clutters (connected antichains) spanning up to n vertices without singleton edges.

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A304984 Number of labeled clutters (connected antichains) spanning some subset of {1,...,n} with singleton edges allowed.

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

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