A341923 -id:A341923 - cp's OEIS frontend

A341856 Array read by antidiagonals: T(n,k) is the number of rooted strong triangulations of a disk with n interior nodes and 3+k nodes on the boundary.

1, 0, 1, 0, 1, 3, 0, 1, 6, 13, 0, 1, 10, 36, 68, 0, 1, 15, 80, 228, 399, 0, 1, 21, 155, 610, 1518, 2530, 0, 1, 28, 273, 1410, 4625, 10530, 16965, 0, 1, 36, 448, 2933, 12165, 35322, 75516, 118668, 0, 1, 45, 696, 5628, 28707, 102548, 272800, 556512, 857956

Offset: 0

Andrew Howroyd, Feb 23 2021

A strong triangulation is one in which no interior edge joins two nodes on the boundary. Except for the single triangle which is enumerated by T(0,0) these are the 3-connected triangulations.

			Array begins:
=======================================================
n\k |    0     1     2      3      4      5       6
----+--------------------------------------------------
  0 |    1     0     0      0      0      0       0 ...
  1 |    1     1     1      1      1      1       1 ...
  2 |    3     6    10     15     21     28      36 ...
  3 |   13    36    80    155    273    448     696 ...
  4 |   68   228   610   1410   2933   5628   10128 ...
  5 |  399  1518  4625  12165  28707  62230  125928 ...
  6 | 2530 10530 35322 102548 267162 638624 1422204 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325
William T. Tutte, A census of planar triangulations, Canad. J. Math. 14 (1962), 21-38.

Columns k=0..3 are A000260, A242136, A341917, A341918.
Antidiagonal sums give A341919.
Cf. A146305 (not necessarily strong triangulations), A210664, A341923, A342053.

T(n,m)=if(m==0, 2*(4*n+1)!/((3*n+2)!*(n+1)!), (3*(m+2)!*(m-1)!/(3*n+3*m+3)!)*sum(j=0, min(m,n-1), (4*n+3*m-j+1)!*(m+j+2)*(m-3*j)/(j!*(j+1)!*(m-j)!*(m-j+2)!*(n-j-1)!)))

T(n,0) = A000260(n) = 2*(4*n+1)!/((3*n+2)!*(n+1)!).

T(n,m) = (3*(m+2)!*(m-1)!/(3*n+3*m+3)!) * Sum_{j=0..min(m,n-1)} (4*n+3*m-j+1)!*(m+j+2)*(m-3*j)/(j!*(j+1)!*(m-j)!*(m-j+2)!*(n-j-1)!) for m > 0.

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.

A378336 Triangle read by rows: T(n,k) is the number of n node connected sensed 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, 5, 5, 6, 5, 0, 0, 2, 8, 13, 20, 21, 26, 24, 0, 0, 0, 10, 28, 55, 79, 104, 119, 147, 133, 0, 0, 0, 7, 45, 126, 230, 360, 491, 625, 735, 892, 846, 0, 0, 0, 0, 44, 227, 561, 1066, 1682, 2430, 3241, 4074, 4830, 5876, 5661

Offset: 3

Andrew Howroyd, Nov 23 2024

See A378103 for illustration of initial terms. This sequence does not consider planar maps to be equivalent to their reflections.
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 up to orientation preserving isomorphisms.
The number of edges is n + k - 1.

			Triangle begins:
n\k | 1  2  3   4   5    6    7    8    9   10   11   12   13
----+----------------------------------------------------------
  3 | 1;
  4 | 0, 1, 1;
  5 | 0, 1, 1,  2,  1;
  6 | 0, 0, 2,  5,  5,   6,   5;
  7 | 0, 0, 2,  8, 13,  20,  21,  26,  24;
  8 | 0, 0, 0, 10, 28,  55,  79, 104, 119, 147, 133;
  9 | 0, 0, 0,  7, 45, 126, 230, 360, 491, 625, 735, 892, 846;
  ...

Andrew Howroyd, Table of n, a(n) for n = 3..2306 (rows 3..50)

Row sums are A378335.
Column sums are A378337.
Antidiagonal sums are A378338.
The final 3 terms of each row are in A002709, A002710, A002711.
Cf. A262586 (2-connected), A341923 (3-connected), A378103, (unsensed), A378340 (achiral).

my(A=A378336rows(10)); for(i=1, #A, print(A[i])) \\ See PARI link in A378340 for program code.

T(n,k) = 0 for n > 2*k + 1.
T(n,2*n-5) = A002709(n-3).
T(n,2*n-6) = A002710(n-4) for n >= 4.
T(n,2*n-7) = A002711(n-5) for n >= 5.

A341925 Number of 3-connected triangulations of a pentagon up to orientation-preserving isomorphisms with n interior nodes.

Original entry on oeis.org

1, 2, 16, 122, 925, 7066, 54560, 426462, 3372291, 26956270, 217614820, 1772645420, 14557912005, 120446430162, 1003264150080, 8408233633984, 70865259181716, 600341163614280, 5109974618238720, 43685248156083450, 374975710962745835, 3230692210606121118

Offset: 1

Andrew Howroyd, Feb 26 2021

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..500

Column k=5 of A341923.

Cf. A342054.

A341926 Number of 3-connected triangulations of a hexagon up to orientation-preserving isomorphisms with n interior nodes.

Original entry on oeis.org

1, 3, 28, 242, 2039, 17138, 142802, 1188412, 9898491, 82665138, 692726420, 5827533378, 49221189621, 417412172800, 3553723666154, 30370156995496, 260485323338444, 2241897876406419, 19358310778238774, 167672728538631224, 1456558329587916711, 12688039161618156484

Offset: 1

Andrew Howroyd, Feb 26 2021

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..500

Column k=6 of A341923.

Cf. A342055.

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.

A341924 Number of 3-connected triangulations of a quadrilateral up to orientation-preserving isomorphisms with n interior nodes.

Original entry on oeis.org

1, 2, 10, 60, 386, 2652, 18914, 139264, 1048947, 8057026, 62893270, 497840520, 3988465676, 32292955800, 263910215250, 2174723058816, 18053970236196, 150883196364072, 1268622177683432, 10725273649776860, 91130026089055954, 777875145348140268, 6667969516814183630

Offset: 1

Andrew Howroyd, Feb 26 2021

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..500

Column k=4 of A341923.

cp's OEIS Frontend

A341856 Array read by antidiagonals: T(n,k) is the number of rooted strong triangulations of a disk with n interior nodes and 3+k nodes on the boundary.

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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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A378336 Triangle read by rows: T(n,k) is the number of n node connected sensed planar maps with an external face and k triangular internal faces, n >= 3, 1 <= k <= 2*n - 5.

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A341925 Number of 3-connected triangulations of a pentagon up to orientation-preserving isomorphisms with n interior nodes.

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A341926 Number of 3-connected triangulations of a hexagon up to orientation-preserving isomorphisms with n interior nodes.

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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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PARI

A341924 Number of 3-connected triangulations of a quadrilateral up to orientation-preserving isomorphisms with n interior nodes.

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