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 1-5 of 5 results.

A263341 Triangle read by rows: T(n,k) is the number of unlabeled simple graphs on n vertices with independence number k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 6, 3, 1, 1, 13, 15, 4, 1, 1, 37, 82, 30, 5, 1, 1, 106, 578, 301, 51, 6, 1, 1, 409, 6021, 4985, 842, 80, 7, 1, 1, 1896, 101267, 142276, 27107, 1995, 117, 8, 1, 1, 12171, 2882460, 7269487, 1724440, 112225, 4210, 164, 9, 1, 1, 105070, 138787233, 655015612, 210799447, 13893557, 388547, 8165, 221, 10, 1
Offset: 1

Views

Author

Christian Stump, Oct 15 2015

Keywords

Comments

The independence number of a graph is the maximum size of an independent set.
Row sums give A000088, n >= 1.
T(n,k) is also the number of graphs on n vertices such that a largest clique is of size k. - Geoffrey Critzer, Sep 23 2016
T(n,k) is also the number of graphs on n vertices such that the size of a smallest vertex cover is n-k. - Geoffrey Critzer, Sep 23 2016
T(n,k) is also the number of graphs on n vertices with independence number k. - Eric W. Weisstein, May 17 2017
For any graph the independence number is greater than or equal to the independent domination number (A332402) and less than or equal to the upper domination number (A332403). - Andrew Howroyd, Feb 19 2020

Examples

			Triangle begins:
  1;
  1,     1;
  1,     2,       1;
  1,     6,       3,       1;
  1,    13,      15,       4,       1;
  1,    37,      82,      30,       5,      1;
  1,   106,     578,     301,      51,      6,    1;
  1,   409,    6021,    4985,     842,     80,    7,   1;
  1,  1896,  101267,  142276,   27107,   1995,  117,   8, 1;
  1, 12171, 2882460, 7269487, 1724440, 112225, 4210, 164, 9, 1;
  ...

Links

Crossrefs

Row sums are A000088.
Columns 2..9 are A052450, A052451, A052452, A077392, A077393, A077394, A205577, A205578.
Transpose of A287024.
Cf. A115196, A126744 (clique number of connected graphs), A294490 (independence number of connected graphs).
Cf. A263284, A332402, A332403.

Extensions

a(21)-a(28) from Geoffrey Critzer, Sep 22 2016
Rows 8-10 from Eric W. Weisstein, May 16 2017
Rows 11-13 from Brendan McKay, Feb 18 2020
Name clarified by Andrew Howroyd, Feb 18 2020

A263284 Triangle read by rows: T(n,k) is the number of unlabeled simple graphs on n vertices with domination number k.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 4, 5, 1, 1, 11, 16, 5, 1, 1, 34, 94, 21, 5, 1, 1, 156, 708, 152, 21, 5, 1, 1, 1044, 9384, 1724, 166, 21, 5, 1, 1, 12346, 221135, 38996, 1997, 166, 21, 5, 1, 1, 274668, 9877969, 1800340, 49961, 2036, 166, 21, 5, 1, 1
Offset: 1

Views

Author

Christian Stump, Oct 13 2015

Keywords

Comments

The domination number of a graph is given by the minimum size of a dominating set of vertices. A dominating set of vertices is a subset of the vertex set of such that every vertex is either in this subset or adjacent to an element of this subset.
For any graph the domination number is greater than or equal to the irredundance number (A332404) and less than or equal to the independent domination number (A332402). - Andrew Howroyd, Feb 13 2020

Examples

			Triangle begins:
       1;
       1,       1;
       2,       1,       1;
       4,       5,       1,     1;
      11,      16,       5,     1,    1;
      34,      94,      21,     5,    1,   1;
     156,     708,     152,    21,    5,   1,  1;
    1044,    9384,    1724,   166,   21,   5,  1, 1;
   12346,  221135,   38996,  1997,  166,  21,  5, 1, 1;
  274668, 9877969, 1800340, 49961, 2036, 166, 21, 5, 1, 1;
  ...

Links

Crossrefs

Row sums are A000088.
Columns k=1..2 are A000088(n-1), A332625.
Cf. A263341, A332400, A332401, A332402, A332403, A332404, A332405.

Formula

T(n,k) = T(n-1,k-1) for 2*(k-1) >= n. - Andrew Howroyd, Feb 17 2020

Extensions

Extended to 10 rows by Eric W. Weisstein, May 18 2017

A332403 Triangle read by rows: T(n,k) is the number of simple graphs on n unlabeled nodes with upper domination number k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 6, 3, 1, 1, 13, 15, 4, 1, 1, 36, 83, 30, 5, 1, 1, 101, 582, 302, 51, 6, 1, 1, 365, 6024, 5025, 843, 80, 7, 1, 1, 1518, 99497, 144370, 27160, 1996, 117, 8, 1, 1, 8002, 2706069, 7441209, 1733211, 112291, 4211, 164, 9, 1
Offset: 1

Views

Author

Andrew Howroyd, Feb 11 2020

Keywords

Comments

First differs from A263341 in row 6.
The upper domination number of a graph is the maximum size of a minimal dominating set (a set that is both dominating and irredundant). For any graph it is greater than or equal to the independence number (A263341) and less than or equal to the upper irredundance number (A332405). The number of graphs where it is strictly greater than is given in A332407.

Examples

			Triangle begins:
  1;
  1,    1;
  1,    2,       1;
  1,    6,       3,       1;
  1,   13,      15,       4,       1;
  1,   36,      83,      30,       5,      1;
  1,  101,     582,     302,      51,      6,    1;
  1,  365,    6024,    5025,     843,     80,    7,   1;
  1, 1518,   99497,  144370,   27160,   1996,  117,   8, 1;
  1, 8002, 2706069, 7441209, 1733211, 112291, 4211, 164, 9, 1;
  ...

Links

Crossrefs

Row sums are A000088.
Cf. A263284, A263341, A332402, A332404, A332405, A332407.

A332404 Triangle read by rows: T(n,k) is the number of simple graphs on n unlabeled nodes with irredundance number k.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 4, 5, 1, 1, 11, 16, 5, 1, 1, 34, 94, 21, 5, 1, 1, 156, 710, 150, 21, 5, 1, 1, 1044, 9419, 1691, 164, 21, 5, 1, 1, 12346, 221979, 38207, 1944, 164, 21, 5, 1, 1, 274668, 9907071, 1773452, 47802, 1983, 164, 21, 5, 1, 1
Offset: 1

Views

Author

Andrew Howroyd, Feb 11 2020

Keywords

Comments

The irredundance number of a graph is the minimum size of a maximal irredundant set.
For any graph the following relation holds:
irredundance number (this sequence)
<= domination number (A263284)
<= independent domination number (A332402)
<= independence number (A263341)
<= upper domination number (A332403)
<= upper irredundance number (A332405).

Examples

			Triangle begins:
       1;
       1,       1;
       2,       1,       1;
       4,       5,       1,     1;
      11,      16,       5,     1,    1;
      34,      94,      21,     5,    1,   1;
     156,     710,     150,    21,    5,   1,  1;
    1044,    9419,    1691,   164,   21,   5,  1, 1;
   12346,  221979,   38207,  1944,  164,  21,  5, 1, 1;
  274668, 9907071, 1773452, 47802, 1983, 164, 21, 5, 1, 1;
  ...

Links

Crossrefs

Row sums are A000088.
Column k=1 is A000088(n-1).
Cf. A263284, A263341, A332402, A332403, A332404, A332405.

Formula

T(n,k) = T(n-1,k-1) for 2*(k-1) >= n.

A332405 Triangle read by rows: T(n,k) is the number of simple graphs on n unlabeled nodes with upper irredundance number k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 6, 3, 1, 1, 13, 15, 4, 1, 1, 36, 83, 30, 5, 1, 1, 101, 582, 302, 51, 6, 1, 1, 364, 6025, 5025, 843, 80, 7, 1, 1, 1511, 99503, 144371, 27160, 1996, 117, 8, 1, 1, 7917, 2706030, 7441332, 1733212, 112291, 4211, 164, 9, 1
Offset: 1

Views

Author

Andrew Howroyd, Feb 11 2020

Keywords

Comments

First differs from A332403 in row 7.
The upper irredundance number of a graph is the maximum size of an irredundant set. For any graph the upper irredundance number is greater than or equal to the upper domination number (A332403).

Examples

			Triangle begins:
  1;
  1,    1;
  1,    2,       1;
  1,    6,       3,       1;
  1,   13,      15,       4,       1;
  1,   36,      83,      30,       5,      1;
  1,  101,     582,     302,      51,      6,    1;
  1,  364,    6025,    5025,     843,     80,    7,   1;
  1, 1511,   99503,  144371,   27160,   1996,  117,   8, 1;
  1, 7917, 2706030, 7441332, 1733212, 112291, 4211, 164, 9, 1;
  ...

Links

Crossrefs

Row sums are A000088.
Cf. A263284, A263341, A332402, A332403, A332404.
Showing 1-5 of 5 results.

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