A378103 -id:A378103 - cp's OEIS frontend

A378340 Triangle read by rows: T(n,k) is the number of n node connected achiral planar maps with an external face and k triangular internal faces, n >= 3, 1 <= k <= 2*n - 5.

1, 0, 1, 1, 0, 1, 1, 2, 1, 0, 0, 2, 3, 3, 4, 3, 0, 0, 2, 4, 7, 8, 7, 10, 8, 0, 0, 0, 4, 8, 15, 19, 22, 19, 29, 23, 0, 0, 0, 3, 11, 22, 32, 48, 57, 65, 57, 86, 68, 0, 0, 0, 0, 8, 25, 47, 82, 104, 150, 175, 200, 176, 266, 215, 0, 0, 0, 0, 7, 26, 64, 123, 186, 288, 346, 488, 556, 634, 557, 844, 680

Offset: 3

Andrew Howroyd, Nov 25 2024

See A378103 for illustration of initial terms. This sequence counts only those maps which have mirror symmetry.
The planar maps considered are without loops or isthmuses.
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, 3,  3,  4,  3;
  7 | 0, 0, 2, 4,  7,  8,  7, 10,  8;
  8 | 0, 0, 0, 4,  8, 15, 19, 22, 19, 29, 23;
  9 | 0, 0, 0, 3, 11, 22, 32, 48, 57, 65, 57, 86, 68;
  ...

Andrew Howroyd, Table of n, a(n) for n = 3..2306 (rows 3..50)
Andrew Howroyd, PARI Program, Nov 2024.

Row sums are A378339.
Column sums are A378341.
Antidiagonal sums are A378342.
Cf. A378103 (unsensed), A378336 (sensed), A002712.

my(A=A378340rows(10)); for(i=1, #A, print(A[i])) \\ See Links for program.

T(n,2*n-5) = A002712(n-3). - Ya-Ping Lu, Dec 16 2024

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

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.

A378343 Number of n edge connected unsensed planar maps with an external face in which all internal faces are triangles.

Original entry on oeis.org

1, 0, 1, 2, 1, 4, 7, 10, 22, 41, 77, 167, 349, 736, 1642, 3627, 8120, 18575, 42548, 97871, 228166, 532484, 1246778, 2941748, 6952421, 16470326, 39217775, 93530974, 223542903, 536138078, 1287943658, 3099775046, 7479610688, 18075710749, 43755189158, 106127886654, 257778445136

Offset: 3

Andrew Howroyd, Nov 25 2024

nonn

Andrew Howroyd, Table of n, a(n) for n = 3..100

Antidiagonal sums of A378103.

Cf. A377785 (by vertices), A378338 (sensed), A378342 (achiral).

A378343seq(30) \\ See PARI link in A378340 for program code.

a(n) = (A378338(n) + A378342(n))/2.

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

Original entry on oeis.org

1, 2, 6, 24, 100, 586, 3725, 26532, 198081, 1539550, 12274565, 99959181, 827795678, 6954099320, 59138955508, 508331799502, 4410651891166, 38590663253312, 340173195849485, 3018768835038348, 26952060900042852, 241960993507098580, 2183134755112963493, 19788571100313277286

Offset: 1

Ya-Ping Lu, Nov 19 2024

nonn

Column sums of A378103.

Cf. A002713, A169808, A377785.

a(n) = (A378337(n) + A378341(n))/2.

cp's OEIS Frontend

A378340 Triangle read by rows: T(n,k) is the number of n node connected achiral planar maps with an external face and k triangular internal faces, n >= 3, 1 <= k <= 2*n - 5.

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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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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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A378343 Number of n edge connected unsensed planar maps with an external face in which all internal faces are triangles.

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PARI

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

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