cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A304997 Number of unlabeled antichains of finite sets spanning n vertices with singleton edges allowed.

Original entry on oeis.org

1, 1, 4, 18, 142, 3100, 823042
Offset: 0

Views

Author

Gus Wiseman, May 23 2018

Keywords

Examples

			Non-isomorphic representatives of the a(3) = 18 antichains:
{{1,2,3}}
{{3},{1,2}}
{{3},{1,2,3}}
{{1,3},{2,3}}
{{1},{2},{3}}
{{2},{3},{1,3}}
{{2},{3},{1,2,3}}
{{3},{1,2},{2,3}}
{{3},{1,3},{2,3}}
{{1,2},{1,3},{2,3}}
{{1},{2},{3},{2,3}}
{{1},{2},{3},{1,2,3}}
{{2},{3},{1,2},{1,3}}
{{2},{3},{1,3},{2,3}}
{{3},{1,2},{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}}

Crossrefs

Cf. A006126, A261005, A304996, A304997, A304998, A304999, A305000, A305001.
See also A374399.

Formula

a(n) = A304996(n) - A304996(n-1) for n > 0. - Andrew Howroyd, Aug 13 2019
Euler transform of A304983. - Andrew Howroyd, Aug 14 2019

Extensions

a(5)-a(6) from Andrew Howroyd, Aug 13 2019

A304985 Number of labeled clutters (connected antichains) spanning n vertices with singleton edges allowed.

Original entry on oeis.org

1, 1, 4, 40, 1344, 203136, 495598592, 309065330371840, 14369391920653644779049472
Offset: 0

Views

Author

Gus Wiseman, May 23 2018

Keywords

Comments

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

Examples

			The a(2) = 4 clutters:
{{1,2}}
{{1},{1,2}}
{{2},{1,2}}
{{1},{2},{1,2}}

Links

Crossrefs

Cf. A048143, A198085, A261006, A304912, A304981, A304982, A304983, A304984, A304985, A304986.
Cf. A000372, A006126, A014466, A304997, A304999, A306505, A317674, A319721.

Formula

For n > 1, a(n) = A048143(n) * 2^n.

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

Views

Author

Gus Wiseman, May 23 2018

Keywords

Comments

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

Examples

			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}}

Crossrefs

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

Formula

Partial sums of A304983.

Extensions

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

Views

Author

Gus Wiseman, May 23 2018

Keywords

Examples

			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}}

Crossrefs

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

Formula

a(n > 0) = A198085(n) + 1.
a(n) = A305005(n) + n.

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

Views

Author

Gus Wiseman, May 23 2018

Keywords

Examples

			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}}

Crossrefs

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

Formula

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

Views

Author

Gus Wiseman, May 23 2018

Keywords

Examples

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

Crossrefs

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

Formula

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

Views

Author

Gus Wiseman, May 23 2018

Keywords

Examples

			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}}

Links

Crossrefs

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

Formula

Binomial transform of A048143 if we assume A048143(1) = 0.
a(n) = A198085(n) - n + 1. - Gus Wiseman, Jun 11 2018
Showing 1-7 of 7 results.

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