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.

A034889 Number of embeddings on the sphere of 2-connected planar graphs with n nodes.

Original entry on oeis.org

1, 3, 10, 61, 564, 7593, 123874, 2262877, 44190279, 904777809, 19207129217, 419870351012, 9405626692325
Offset: 3

Views

Author

Ronald C. Read

Keywords

Comments

The complete graph on two vertices is sometimes considered to be 2-connected (or nonseparable). Compare A002218 with A021103. - Andrew Howroyd, Mar 01 2021

References

  • R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.

Links

  • G. Brinkmann and B. D. McKay, Fast generation of planar graphs, MATCH Commun. Math. Comput. Chem, 58 (2007), 323-357. [Expanded version]
  • Eric Weisstein's World of Mathematics, Planar Embedding.

Crossrefs

Row sums of A342060.
Cf. A002218, A005142, A021103.

Extensions

a(8)-a(15) added by Mohammadreza Jooyandeh, Sep 03 2013

A384963 Triangle read by rows: T(n,k) is the number of embeddings on the sphere of connected simple planar graphs with n nodes and k faces, n >= 1, k=1..max(1,2*n-4).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 1, 3, 7, 7, 5, 2, 1, 6, 22, 42, 49, 35, 18, 5, 2, 12, 76, 237, 442, 510, 412, 218, 84, 18, 5, 27, 271, 1293, 3539, 6205, 7482, 6318, 3833, 1623, 485, 88, 14, 65, 1001, 6757, 25842, 63254, 106985, 129782, 115988, 76582, 37421, 13111, 3228, 489, 50
Offset: 1

Views

Author

Andrew Howroyd, Jun 13 2025

Keywords

Comments

Equivalently, T(n,k) is the number of unsensed simple planar maps with n vertices and k faces.
The number of edges is n+k-2.
Terms of this sequence can be computed using the tool "plantri". The expanded reference gives rows 1..14 of this table.

Examples

			Triangle begins:
   1;
   1;
   1,   1;
   2,   2,    1,    1,
   3,   7,    7,    5,    2,    1;
   6,  22,   42,   49,   35,   18,    5,    2;
  12,  76,  237,  442,  510,  412,  218,   84,   18,   5;
  27, 271, 1293, 3539, 6205, 7482, 6318, 3833, 1623, 485, 88, 14;
  ...

Links

Crossrefs

Row sums are A372892.
Antidiagonal sums are A006395.
Columns 1..2 are A006082, A384967.
Cf. A277741 (not necessarily simple), A342060 (2-connected), A212438 (3-connected), A384850 (version by number of edges then vertices), A384964 (sensed version).

A342059 Triangle read by rows: T(n,k) is the number of embeddings on the sphere of 2-connected planar graphs with n nodes and k faces up to orientation preserving isomorphisms, n >= 3, k=2..2*n-4.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 5, 2, 1, 1, 3, 17, 31, 22, 6, 2, 1, 4, 42, 157, 318, 265, 123, 26, 6, 1, 6, 87, 576, 2128, 4009, 4055, 2332, 804, 147, 17, 1, 7, 161, 1664, 9659, 31252, 59244, 66289, 46521, 20604, 5743, 892, 73, 1, 9, 286, 4151, 34700, 168757, 505410, 952044, 1156127, 931227, 506318, 183980, 43180, 5876, 389
Offset: 3

Views

Author

Andrew Howroyd, Mar 27 2021

Keywords

Comments

The number of edges is n+k-2.
Terms of this sequence can be computed using the tool "plantri". The expanded reference gives rows 3..15 of this table.

Examples

			Triangle begins:
  1;
  1, 1,  1;
  1, 2,  5,   2,    1;
  1, 3, 17,  31,   22,    6,    2;
  1, 4, 42, 157,  318,  265,  123,   26,   6;
  1, 6, 87, 576, 2128, 4009, 4055, 2332, 804, 147, 17;
  ...

Links

Crossrefs

Row sums are A342058.
Cf. A006406 (by edges), A239893 (3-connected), A342060.

Formula

T(n,2) = 1.
T(n,3) = A253186(n-2).

A006407 Number of unsensed 2-connected simple planar maps with n edges.

Original entry on oeis.org

1, 1, 2, 4, 8, 20, 58, 177, 630, 2410, 9772, 41423, 181586, 814412, 3722445
Offset: 3

Views

Author

N. J. A. Sloane

Keywords

Comments

A simple planar map is a planar map without loops or parallel edges.
Equivalently, a(n) is the number of embeddings on the sphere of 2-connected planar graphs with n edges. - Andrew Howroyd, Mar 27 2021

References

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

Links

Crossrefs

Cf. A006403, A006405, A006406 (sensed case), A342060, A379437 (rooted).

Formula

a(n) = Sum_{k=3..n} A342060(k, n+2-k). - Andrew Howroyd, Mar 27 2021

Extensions

a(11) and a(12) from Sean A. Irvine, Apr 03 2017
a(13)-a(17) from Andrew Howroyd, Mar 27 2021

A378075 Triangle read by rows: T(n,k) is the number of embeddings on the sphere of 2-connected homeomorphically irreducible planar graphs with n nodes and k faces, k=4..2n-4.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 3, 3, 2, 0, 0, 3, 11, 18, 10, 5, 0, 0, 3, 19, 77, 134, 123, 50, 14, 0, 0, 0, 13, 146, 603, 1280, 1420, 883, 278, 50, 0, 0, 0, 8, 162, 1409, 6030, 13781, 18404, 14570, 6884, 1772, 233, 0, 0, 0, 0, 83, 1809, 15225, 64502, 158717, 240841, 233286, 144005, 55444, 12077, 1249
Offset: 4

Views

Author

Andrew Howroyd, Nov 15 2024

Keywords

Comments

The number of edges is n + k - 2.

Examples

			Triangle begins:
  n\k| 4  5  6   7    8     9    10     11     12     13    14    15   16
-----+--------------------------------------------------------------------
   4 | 1;
   5 | 0, 1, 1;
   6 | 0, 1, 3,  3,   2;
   7 | 0, 0, 3, 11,  18,   10,    5;
   8 | 0, 0, 3, 19,  77,  134,  123,    50,    14;
   9 | 0, 0, 0, 13, 146,  603, 1280,  1420,   883,   278,   50;
  10 | 0, 0, 0,  8, 162, 1409, 6030, 13781, 18404, 14570, 6884, 1772, 233;
  ...

Links

Crossrefs

Row sums are A378074.
Antidiagonal sums give A378076.
Cf. A212438, A342060, A378077.

Formula

T(n,k) = A212438(n,k) + A378077(n,k).

A384850 Triangle read by rows: T(n,k) is the number of unsensed simple planar maps with n edges and k vertices, 1 <= k <= n+1.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 2, 3, 0, 0, 0, 1, 7, 6, 0, 0, 0, 1, 7, 22, 12, 0, 0, 0, 0, 5, 42, 76, 27, 0, 0, 0, 0, 2, 49, 237, 271, 65, 0, 0, 0, 0, 1, 35, 442, 1293, 1001, 175, 0, 0, 0, 0, 0, 18, 510, 3539, 6757, 3765, 490
Offset: 0

Views

Author

Andrew Howroyd, Jun 13 2025

Keywords

Comments

The planar maps considered here are connected.
The initial terms of this sequence can be computed using the tool "plantri", in particular the command "./plantri -u -v -c1 -p [n]" will compute values for a column.

Examples

			Triangle begins:
  1;
  0, 1;
  0, 0, 1;
  0, 0, 1, 2;
  0, 0, 0, 2, 3;
  0, 0, 0, 1, 7,  6;
  0, 0, 0, 1, 7, 22,  12;
  0, 0, 0, 0, 5, 42,  76,   27;
  0, 0, 0, 0, 2, 49, 237,  271,   65;
  0, 0, 0, 0, 1, 35, 442, 1293, 1001, 175;
  ...

Links

Crossrefs

Row sums are A006395.
Column sums are A372892.
Main diagonal is A006082.
Subdiagonal is A384967.
Cf. A054923 (graphs), A277741 (not necessarily simple), A342060 (2-connected), A212438 (3-connected), A384963 (version by number of vertices then faces).

A379432 Triangle read by rows: T(n,k) is the number of unsensed 2-connected (nonseparable) planar maps with n edges and k vertices, n >= 2, 2 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 7, 3, 1, 1, 4, 13, 13, 4, 1, 1, 5, 29, 44, 29, 5, 1, 1, 7, 51, 139, 139, 51, 7, 1, 1, 8, 92, 370, 623, 370, 92, 8, 1, 1, 10, 147, 913, 2307, 2307, 913, 147, 10, 1, 1, 12, 240, 2048, 7644, 11673, 7644, 2048, 240, 12, 1, 1, 14, 357, 4295, 22344, 50174, 50174, 22344, 4295, 357, 14, 1
Offset: 2

Views

Author

Andrew Howroyd, Jan 14 2025

Keywords

Comments

The maps considered here may include parallel edges.
The number of faces is n + 2 - k.

Examples

			Triangle begins:
   1;
   1,  1;
   1,  1,   1;
   1,  2,   2,   1;
   1,  3,   7,   3,    1;
   1,  4,  13,  13,    4,    1;
   1,  5,  29,  44,   29,    5,   1;
   1,  7,  51, 139,  139,   51,   7,   1;
   1,  8,  92, 370,  623,  370,  92,   8,  1;
   1, 10, 147, 913, 2307, 2307, 913, 147, 10, 1;
   ...

Links

Crossrefs

Row sums are A006403.
Cf. A082680 (rooted), A342061 (sensed), A212438 (3-connected), A277741, A342060.

Formula

T(n,k) = T(n, n+2-k).
Showing 1-7 of 7 results.

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

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