A002713 -id:A002713 - cp's OEIS frontend

A169808 Array T(n,k) read by antidiagonals: T(n,k) is the number of [n,k]-triangulations in the plane, n >= 0, k >= 0.

1, 1, 1, 1, 2, 1, 3, 4, 5, 4, 4, 11, 14, 18, 16, 12, 28, 53, 69, 88, 78, 27, 91, 178, 295, 396, 489, 457, 82, 291, 685, 1196, 1867, 2503, 3071, 2938, 228, 1004, 2548, 5051, 8385, 12560, 16905, 20667, 20118, 733, 3471, 9876, 21018, 38078, 60736, 89038, 119571, 146381, 144113

Offset: 0

N. J. A. Sloane, May 25 2010

"A closed bounded region in the plane divided into triangular regions with k+3 vertices on the boundary and n internal vertices is said to be a triangular map of type [n,k]." It is a [n,k]-triangulation if there are no multiple edges.
T(n,k) is the number of floor plan arrangements represented by 3-connected trivalent maps with n internal rooms and k+3 rooms adjacent to the outside.
"... may be evaluated from the results given by Brown."
The initial terms of this sequence can also be computed using the tool "plantri", in particular the command "./plantri -u -v -P -c2m2 [n]" will compute values for a diagonal. The '-c2' and '-m2' options indicate graphs must be biconnected and with minimum vertex degree 2. - Andrew Howroyd, Feb 22 2021

			Array begins:
============================================================
n\k |    0     1      2      3       4        5        6
----+-------------------------------------------------------
  0 |    1     1      1      3       4       12       27 ...
  1 |    1     2      4     11      28       91      291 ...
  2 |    1     5     14     53     178      685     2548 ...
  3 |    4    18     69    295    1196     5051    21018 ...
  4 |   16    88    396   1867    8385    38078   169918 ...
  5 |   78   489   2503  12560   60736   290595  1367374 ...
  6 |  457  3071  16905  89038  451613  2251035 11025626 ...
  7 | 2938 20667 119571 652198 3429943 17658448 89328186 ...
  ...

C. F. Earl and L. J. March, Architectural applications of graph theory, pp. 327-355 of R. J. Wilson and L. W. Beineke, editors, Applications of Graph Theory. Academic Press, NY, 1979.

Andrew Howroyd, Table of n, a(n) for n = 0..1325
G. Brinkmann and B. McKay, Plantri (program for generation of certain types of planar graph)
William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768.
William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768. [Annotated scanned copy].
C. F. Earl and L. J. March, Architectural applications of graph theory, pp. 327-355 of R. J. Wilson and L. W. Beineke, editors, Applications of Graph Theory. Academic Press, NY, 1979. (Annotated scanned copy)
Andrew Howroyd, PARI Program

Columns k=0..3 are A002713, A005500, A005501, A005502.
Rows n=0..2 are A000207, A005503, A005504.
Antidiagonal sums give A005027.
Cf. A146305 (rooted), A169809 (achiral), A262586 (oriented).

\\ See link for script file.
A169808Array(6) \\ Andrew Howroyd, Feb 22 2021

T(n,k) = (A262586(n,k) + A169809(n,k)) / 2. - Andrew Howroyd, Feb 22 2021

Edited by Andrew Howroyd, Feb 22 2021

a(29) corrected and terms a(36) and beyond from Andrew Howroyd, Feb 22 2021

A342053 Array read by antidiagonals: T(n,k) is the number of unrooted 3-connected triangulations of a disk with n interior nodes and k nodes on the boundary, n >= 1, k >= 3.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 1, 2, 8, 16, 1, 3, 12, 38, 78, 1, 3, 20, 73, 219, 457, 1, 4, 27, 140, 503, 1404, 2938, 1, 4, 39, 235, 1089, 3661, 9714, 20118, 1, 5, 51, 392, 2149, 8796, 27715, 70454, 144113, 1, 5, 68, 610, 4050, 19419, 72204, 214664, 527235, 1065328

Offset: 1

Andrew Howroyd, Feb 26 2021

For k >= 4, T(n,k) is the number of polyhedra with n+k vertices whose faces are all triangular, except one which is k-gonal.

The initial terms of this sequence can also be computed using the tool "plantri", in particular the command "./plantri -u -v -P [n]" will compute values for a diagonal.

			Array begins:
===================================================
n\k |     3     4      5      6       7       8
----+----------------------------------------------
  1 |     1     1      1      1       1       1 ...
  2 |     1     2      2      3       3       4 ...
  3 |     4     8     12     20      27      39 ...
  4 |    16    38     73    140     235     392 ...
  5 |    78   219    503   1089    2149    4050 ...
  6 |   457  1404   3661   8796   19419   40485 ....
  7 |  2938  9714  27715  72204  173779  393123 ...
  8 | 20118 70454 214664 596906 1538221 3723976 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275
G. Brinkmann and B. McKay, Plantri (program for generation of certain types of planar graph)
William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768.
Andrew Howroyd, PARI Program

Columns k=3..6 are A002713, A058786(n+4), A342054, A342055.
Antidiagonal sums are A342056.
Cf. A169808 (2-connected), A341856 (rooted), A341923 (oriented).

A342053Array(8,6) \\ See links for program.

A378103 Triangle read by rows: T(n,k) is the number of n-node connected unsensed planar maps with an external face and k triangular internal faces, n >= 3, 1 <= k <= 2*n - 5.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 2, 1, 0, 0, 2, 4, 4, 5, 4, 0, 0, 2, 6, 10, 14, 14, 18, 16, 0, 0, 0, 7, 18, 35, 49, 63, 69, 88, 78, 0, 0, 0, 5, 28, 74, 131, 204, 274, 345, 396, 489, 457, 0, 0, 0, 0, 26, 126, 304, 574, 893, 1290, 1708, 2137, 2503, 3071, 2938, 0, 0, 0, 0, 13, 159, 582, 1396, 2613, 4274, 6270, 8709, 11433, 14227, 16905, 20667, 20118

Offset: 3

Ya-Ping Lu, Nov 16 2024

The planar maps considered are without loops or isthmuses.
In other words, a(n) is the number of embeddings in the plane of connected bridgeless planar simple graphs with n vertices and k triangular internal faces.
The number of edges is n + k - 1.
The nonzero terms in row n range from k = floor(n/2) through 2*n-5 and, thus, the number of nonzero terms is 2n - floor(n/2) - 4 = A001651(n-2).

			Triangle begins:
n\k        1     2     3     4     5     6     7     8     9    10    11
----     ----  ----  ----  ----  ----  ----  ----  ----  ----  ----  ----
3          1
4          0     1     1
5          0     1     1     2     1
6          0     0     2     4     4     5     4
7          0     0     2     6    10    14    14    18    16
8          0     0     0     7    18    35    49    63    69    88    78

Andrew Howroyd, Table of n, a(n) for n = 3..2306 (rows 3..50)
Ya-Ping Lu, Illustration of initial terms

Row sums are A377785.
Cf. A001651, A002713, A003094, A169808, A378336 (sensed), A378340 (achiral).
The final 3 terms of each row are in A002713, A005500, A005501.

my(A=A378103rows(10)); for(i=1, #A, print(A[i])) \\ See PARI link in A378340 for program code. - Andrew Howroyd, Nov 25 2024

T(n, 2*n-5) = A002713(n-3).

T(n,k) = (A378336(n,k) + A378340(n,k))/2.

a(39) onwards from Andrew Howroyd, Nov 25 2024

A262322 The number of 4-connected triangulations of the triangle with n inner vertices.

Original entry on oeis.org

1, 0, 0, 1, 1, 3, 13, 47, 217, 1041, 5288, 27844, 150608, 831229

Offset: 0

Moritz Firsching, Sep 18 2015

Also the number of 4-connected simplicial polyhedra with n nodes with one marked face.

Values obtained by generating 4-connected simplicial polyhedra with plantri, marking each face in the polyhedron, and then sorting out isomorphic ones.

Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph.
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph [Cached copy, pdf file only, no active links, with permission]

Cf. A002713, A007021.

cp's OEIS Frontend

A169808 Array T(n,k) read by antidiagonals: T(n,k) is the number of [n,k]-triangulations in the plane, n >= 0, k >= 0.

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A342053 Array read by antidiagonals: T(n,k) is the number of unrooted 3-connected triangulations of a disk with n interior nodes and k nodes on the boundary, n >= 1, k >= 3.

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A378103 Triangle read by rows: T(n,k) is the number of n-node connected unsensed planar maps with an external face and k triangular internal faces, n >= 3, 1 <= k <= 2*n - 5.

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A262322 The number of 4-connected triangulations of the triangle with n inner vertices.

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A378190 Number of planar maps with an external face and n internal triangular faces.

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