A384967 -id:A384967 - cp's OEIS frontend

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).

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

Andrew Howroyd, Jun 13 2025

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.

			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;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..158 (rows 1..14)
Gunnar Brinkmann and Brendan McKay, Fast generation of planar graphs (expanded edition), Tables 19-22.

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).

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

Andrew Howroyd, Jun 13 2025

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.

			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;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..135 (rows 0..15)
Gunnar Brinkmann and Brendan McKay, Plantri (program for generation of certain types of planar graph).

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).

A384966 Number of sensed simple planar maps with n vertices and 2 faces.

Original entry on oeis.org

0, 0, 1, 2, 8, 29, 113, 444, 1763, 6951, 27395, 107672, 422330, 1654180, 6472518, 25308760, 98923442, 386589398, 1510737079, 5904291401, 23079308104, 90236258057, 352908128341, 1380632536468, 5403055984114, 21152009997924, 82835786189975, 324518950873991, 1271797441923614, 4985982054721119

Offset: 1

Andrew Howroyd, Jun 14 2025

nonn

In other words, a(n) is the number of embeddings on the sphere of connected simple unicyclic planar graphs with n nodes up to orientation preserving isomorphisms.

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Column 2 of A384964.

Cf. A001429, A006078 (cycle is loop), A007595 (cycle is digon), A380237 (not necessarily simple), A384967 (unsensed version)..

seq(n)={my(c(d)=(1-sqrt(1-4*x^d + O(x*x^(n+d))))/(2*x^d)); Vec(1/(1 - x*c(2)) - x*(c(1)^2 + c(2)) - x^2*(c(1)^4 + 3*c(2)^2)/2 - 1 - sum(k=1, n, log(2 - c(k))*eulerphi(k)/k), -n)/2}

a(n) = A380237(n) - A007595(n) - A006078(n).

cp's OEIS Frontend

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).

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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.

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A384966 Number of sensed simple planar maps with n vertices and 2 faces.

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