A342056 -id:A342056 - cp's OEIS frontend

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.

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.

A377785 Number of n-node connected unsensed planar maps with an external face in which all internal faces are triangles.

Original entry on oeis.org

1, 2, 5, 19, 80, 407, 2403, 15570, 107366, 773660, 5746826, 43693041, 338367370, 2660629311, 21193182914, 170714892489, 1388711749891, 11395477091396, 94238431709661, 784796579069288, 6576997589788217, 55435506835745851, 469698698872829672, 3998806331800289798

Offset: 3

Ya-Ping Lu, Nov 06 2024

nonn

Maps for a(3) through a(7) are illustrated in Links.
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 in which each internal face is a triangle.

Andrew Howroyd, Table of n, a(n) for n = 3..200
Ya-Ping Lu, Illustration of initial terms

Row sums of A378103.

Cf. A005027 (2-connected), A342056 (3-connected), A378335 (sensed), A378339 (achiral), A378340.

A377785seq(30) \\ See PARI link in A378340 for program code.

a(n) = (A378335(n) + A378339(n))/2. - Andrew Howroyd, Nov 24 2024

a(9) onwards from Andrew Howroyd, Nov 24 2024

A342052 Number of 3-connected triangulations of a disk with n nodes up to orientation-preserving isomorphisms.

Original entry on oeis.org

1, 2, 8, 37, 213, 1386, 9524, 68057, 501858, 3788747, 29170667, 228295618, 1811802818, 14552804492, 118124257451, 967698049455, 7992746427963, 66500865364037, 556944249243331, 4692174542007030, 39744552170122779, 338311257783873501, 2892751486485359650

Offset: 4

Andrew Howroyd, Feb 26 2021

nonn

Andrew Howroyd, Table of n, a(n) for n = 4..100
G. Brinkmann and B. D. McKay, Fast generation of planar graphs (expanded edition), Table 27, Column d'1.

Antidiagonal sums of A341923.

Cf. A342053, A342056.

A341923AntidiagonalSums(25) \\ See links in A342053 for program.

cp's OEIS Frontend

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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A377785 Number of n-node connected unsensed planar maps with an external face in which all internal faces are triangles.

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A342052 Number of 3-connected triangulations of a disk with n nodes up to orientation-preserving isomorphisms.

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Author

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Links

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Programs

PARI