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.

Previous Showing 31-40 of 83 results. Next

A326906 Number of sets of subsets of {1..n} that are closed under union and cover all n vertices.

Original entry on oeis.org

2, 2, 8, 90, 4542, 2747402, 151930948472, 28175295407840207894
Offset: 0

Views

Author

Gus Wiseman, Aug 03 2019

Keywords

Comments

Differs from A102895 in having a(0) = 2 instead of 1.

Examples

			The a(0) = 2 through a(2) = 8 sets of subsets:
  {}    {{1}}     {{1,2}}
  {{}}  {{},{1}}  {{},{1,2}}
                  {{1},{1,2}}
                  {{2},{1,2}}
                  {{},{1},{1,2}}
                  {{},{2},{1,2}}
                  {{1},{2},{1,2}}
                  {{},{1},{2},{1,2}}

Crossrefs

The case without empty sets is A102894.
The case with a single covering edge is A102895.
Binomial transform is A102897.
The case also closed under intersection is A326878 for n > 0.
The same for intersection instead of union is (also) A326906.
The unlabeled version is A326907.
Cf. A000798, A102896, A102897, A108800, A193675, A306445, A326880, A326881, A326883.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Subsets[Range[n]]],Union@@#==Range[n]&&SubsetQ[#,Union@@@Tuples[#,2]]&]],{n,0,3}]

Formula

a(n) = 2 * A102894(n).

A326883 Number of unlabeled set-systems with {} that are closed under intersection and cover n vertices.

Original entry on oeis.org

1, 1, 4, 22, 302, 28630, 216533404, 5592325966377736
Offset: 0

Views

Author

Gus Wiseman, Jul 30 2019

Keywords

Examples

			Non-isomorphic representatives of the a(0) = 1 through a(3) = 22 set-systems:
  {{}}  {{}{1}}  {{}{12}}        {{}{123}}
                 {{}{1}{2}}      {{}{1}{23}}
                 {{}{2}{12}}     {{}{3}{123}}
                 {{}{1}{2}{12}}  {{}{1}{2}{3}}
                                 {{}{23}{123}}
                                 {{}{1}{3}{23}}
                                 {{}{2}{3}{123}}
                                 {{}{3}{13}{23}}
                                 {{}{1}{23}{123}}
                                 {{}{3}{23}{123}}
                                 {{}{1}{2}{3}{23}}
                                 {{}{1}{2}{3}{123}}
                                 {{}{2}{3}{13}{23}}
                                 {{}{1}{3}{23}{123}}
                                 {{}{2}{3}{23}{123}}
                                 {{}{3}{13}{23}{123}}
                                 {{}{1}{2}{3}{13}{23}}
                                 {{}{1}{2}{3}{23}{123}}
                                 {{}{2}{3}{13}{23}{123}}
                                 {{}{1}{2}{3}{12}{13}{23}}
                                 {{}{1}{2}{3}{13}{23}{123}}
                                 {{}{1}{2}{3}{12}{13}{23}{123}}

Crossrefs

The case also closed under union is A001930.
The connected case (i.e., with maximum) is A108798.
The same for union instead of intersection is (also) A108798.
The non-covering case is A108800.
The labeled case is A326881.
Cf. A000798, A102894, A102895, A102897, A193674, A193675, A306445, A326875, A326878, A326880.

Formula

a(n) = A108800(n) - A108800(n-1) for n > 0. - Andrew Howroyd, Aug 10 2019

Extensions

a(5)-a(7) from Andrew Howroyd, Aug 10 2019

A326901 Number of set-systems (without {}) on n vertices that are closed under intersection.

Original entry on oeis.org

1, 2, 6, 32, 418, 23702, 16554476, 1063574497050, 225402367516942398102
Offset: 0

Views

Author

Gus Wiseman, Aug 04 2019

Keywords

Comments

A set-system is a finite set of finite nonempty sets, so no two edges of a set-system that is closed under intersection can be disjoint.

Examples

			The a(3) = 32 set-systems:
  {}  {{1}}    {{1}{12}}    {{1}{12}{13}}   {{1}{12}{13}{123}}
      {{2}}    {{1}{13}}    {{2}{12}{23}}   {{2}{12}{23}{123}}
      {{3}}    {{2}{12}}    {{3}{13}{23}}   {{3}{13}{23}{123}}
      {{12}}   {{2}{23}}    {{1}{12}{123}}
      {{13}}   {{3}{13}}    {{1}{13}{123}}
      {{23}}   {{3}{23}}    {{2}{12}{123}}
      {{123}}  {{1}{123}}   {{2}{23}{123}}
               {{2}{123}}   {{3}{13}{123}}
               {{3}{123}}   {{3}{23}{123}}
               {{12}{123}}
               {{13}{123}}
               {{23}{123}}

Links

Crossrefs

The case with union instead of intersection is A102896.
The case closed under union and intersection is A326900.
The covering case is A326902.
The connected case is A326903.
The unlabeled version is A326904.
The BII-numbers of these set-systems are A326905.
Cf. A000798, A001930, A006058, A102895, A102898, A182507, A326866, A326876, A326878, A326882.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}]

Formula

a(n) = 1 + Sum_{k=0, n-1} binomial(n,k)*A102895(k). - Andrew Howroyd, Aug 10 2019

Extensions

a(5)-a(8) from Andrew Howroyd, Aug 10 2019

A001827 Related to graded partially ordered sets.

Original entry on oeis.org

1, 4, 22, 166, 1726, 24814, 494902, 13729846, 531077086, 28697950174, 2170176736102, 230007989092006, 34211282155446286, 7149766552058591374, 2101690590380890192342, 869808621195903097079446, 507261036269544624540347326
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Corresponds to the numbers c(4,n) in the Klarner paper. - Sean A. Irvine, Sep 24 2015

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Column k=4 of A361950.
Cf. A000684, A001828, A001829, A001830, A047863.

Formula

a(n) = Sum_{p+q+r+s=n} (n!/p!q!r!s!) 2^(pq+qr+rs) where (p,q,r,s) is any nonnegative composition of n. - Sean A. Irvine, Sep 24 2015

Extensions

More terms from Sean A. Irvine, Sep 24 2015

A001828 Related to graded partially ordered sets.

Original entry on oeis.org

1, 5, 33, 287, 3309, 50975, 1058493, 29885567, 1156711869, 61815727295, 4589058616413, 475576073939807, 69061902766811229, 14093318360697120095, 4049931601653596366013, 1641314561238334948886207, 939097032426474389539281789
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Corresponds to the numbers c(5,n) in the Klarner paper. - Sean A. Irvine, Sep 24 2015

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Column k=5 of A361950.
Cf. A000684, A001827, A001829, A001830, A047863.

Formula

a(n) = Sum_{p+q+r+s+t=n} (n!/p!q!r!s!t!) 2^(pq+qr+rs+st) where (p,q,r,s,t) is any nonnegative composition of n. - Sean A. Irvine, Sep 24 2015

Extensions

More terms from Sean A. Irvine, Sep 24 2015

A001829 Related to graded partially ordered sets.

Original entry on oeis.org

1, 6, 46, 450, 5650, 91866, 1957066, 55363650, 2109599650, 109773407466, 7894945079386, 792252362302770, 111671194813402930, 22202849561274787866, 6241728810901739517226, 2484011055161613143144610, 1400187830319472451472442690
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Corresponds to the numbers c(6,n) in the Klarner paper. - Sean A. Irvine, Sep 24 2015

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Column k=6 of A361950.
Cf. A000684, A001827, A001828, A001830, A047863.

Formula

a(n) = Sum_{p+q+r+s+t+u=n} (n!/p!q!r!s!t!u!) 2^(pq+qr+rs+st+tu) where (p,q,r,s,t,u) is any nonnegative composition of n. - Sean A. Irvine, Sep 24 2015

A001830 Related to graded partially ordered sets.

Original entry on oeis.org

1, 7, 61, 661, 8953, 152917, 3334921, 94354981, 3528929353, 177999003157, 12340001650921, 1194005625114661, 162936187792764073, 31536761103831315157, 8677703806537883683081, 3395880602480076153665701, 1889190751946097573211698313
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Corresponds to the numbers c(7,n) in the Klarner paper. - Sean A. Irvine, Sep 24 2015

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Column k=7 of A361950.
Cf. A000684, A001827, A001828, A001829, A047863.

Formula

a(n) = Sum_{p+q+r+s+t+u+v=n} (n!/p!q!r!s!t!u!v!) 2^(pq+qr+rs+st+tu+uv) where (p,q,r,s,t,u,v) is any nonnegative composition of n. - Sean A. Irvine, Sep 24 2015

Extensions

More terms from Sean A. Irvine, Sep 24 2015

A326898 Number of unlabeled topologies with up to n points.

Original entry on oeis.org

1, 2, 5, 14, 47, 186, 904, 5439, 41418, 404501, 5122188, 84623842, 1828876351, 51701216248, 1908493827243, 91755916071736, 5729050033597431
Offset: 0

Views

Author

Gus Wiseman, Aug 02 2019

Keywords

Examples

			Non-isomorphic representatives of the a(0) = 1 through a(3) = 14 topologies:
  {}  {}     {}            {}
      {}{1}  {}{1}         {}{1}
             {}{12}        {}{12}
             {}{2}{12}     {}{123}
             {}{1}{2}{12}  {}{2}{12}
                           {}{3}{123}
                           {}{23}{123}
                           {}{1}{2}{12}
                           {}{1}{23}{123}
                           {}{3}{23}{123}
                           {}{2}{3}{23}{123}
                           {}{3}{13}{23}{123}
                           {}{2}{3}{13}{23}{123}
                           {}{1}{2}{3}{12}{13}{23}{123}

Crossrefs

Partial sums of A001930.
The labeled version is A326878.
Cf. A000612, A000798, A003180, A108800, A193675, A306445, A326867, A326882.

A326904 Number of unlabeled set-systems (without {}) on n vertices that are closed under intersection.

Original entry on oeis.org

1, 2, 4, 10, 38, 368, 29328, 216591692, 5592326399531792
Offset: 0

Views

Author

Gus Wiseman, Aug 04 2019

Keywords

Comments

A set-system is a finite set of finite nonempty sets, so no two edges of such a set-system can be disjoint.
Apart from the offset the same as A193675. - R. J. Mathar, Aug 09 2019

Examples

			Non-isomorphic representatives of the a(0) = 1 through a(3) = 10 set-systems:
  {}  {}     {}           {}
      {{1}}  {{1}}        {{1}}
             {{1,2}}      {{1,2}}
             {{2},{1,2}}  {{1,2,3}}
                          {{2},{1,2}}
                          {{3},{1,2,3}}
                          {{2,3},{1,2,3}}
                          {{3},{1,3},{2,3}}
                          {{3},{2,3},{1,2,3}}
                          {{3},{1,3},{2,3},{1,2,3}}

Crossrefs

The covering case is A108800(n - 1).
The case with an edge containing all of the vertices is A193674(n - 1).
The case with union instead of intersection is A193674.
The labeled version is A326901.
Cf. A000798, A001930, A006058, A102895, A102898, A326876, A326866, A326878, A326882, A326903, A326906.

Formula

a(n > 0) = 2 * A193674(n - 1).

A003431 Number of isomorphism classes of connected irreducible posets with n labeled points.

Original entry on oeis.org

1, 1, 0, 0, 1, 12, 104, 956, 10037, 126578, 1971005, 38569954, 958347642, 30400603560, 1234260982770, 64187360439352, 4275470549123119
Offset: 0

Views

Author

N. J. A. Sloane and Richard Stanley

Keywords

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A003430, A046906.

Extensions

More terms from Richard Stanley, Jun 19 2003
2 more terms from Vladeta Jovovic, Feb 17 2008
Title clarified by Geoffrey Critzer, Jul 08 2022
a(0) changed to 1 by Geoffrey Critzer, Jul 10 2022
Previous Showing 31-40 of 83 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.