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.

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

Author

Andrew Howroyd, Nov 23 2024

Keywords

• nonn
• tabf

Comments

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.

Examples

			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;
  ...
		

Links

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

Crossrefs

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).

Programs

PARI

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

Formula

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.