A049334 Triangle read by rows: T(n, k) is the number of unlabeled connected planar simple graphs with n >= 1 nodes and 0<=k<=3*n-6 edges.

1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 2, 1, 1, 0, 0, 0, 0, 3, 5, 5, 4, 2, 1, 0, 0, 0, 0, 0, 6, 13, 19, 22, 19, 13, 5, 2, 0, 0, 0, 0, 0, 0, 11, 33, 67, 107, 130, 130, 96, 51, 16, 5, 0, 0, 0, 0, 0, 0, 0, 23, 89, 236, 486, 804, 1112, 1211, 1026, 626, 275, 72, 14, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Brendan McKay

Keywords

• nonn
• tabf
• nice

Comments

Planar graphs with n >= 3 nodes have at most 3*n-6 edges.

Examples

			n\k 0  1  2  3  4  5  6  7  8  9 10 11 12
--:-- -- -- -- -- -- -- -- -- -- -- -- --
1:  1
2:  0  1
3:  0  0  1  1
4:  0  0  0  2  2  1  1
5:  0  0  0  0  3  5  5  4  2  1
6:  0  0  0  0  0  6 13 19 22 19 13  5  2
		

Links

• Georg Grasegger, Table of n, a(n) for n = 1..235 (rows 1..13) (terms n = 1..147 (rows 1..11) from Andrew Howroyd)
• F. Harary, The number of linear, directed, rooted, and connected graphs, Trans. Amer. Math. Soc. 78 (1955), 445-463. (MR0068198) See page 457, equation (2.9).

Crossrefs

Row sums are A003094.

Column sums are A046091.

Cf. A000055, A039735, A049336, A049337, A054924, A288265, A343873 (transpose).

Programs

nauty

geng -c $n $k:$k | planarg -q | countg -q # Georg Grasegger, Jul 11 2023

Formula

T(n, n-1) = A000055(n) and Sum_{k} T(n, k) = A003094(n) if n>=1. - Michael Somos, Aug 23 2015

log(1 + B(x, y)) = Sum{n>0} A(x^n, y^n) / n where A(x, y) = Sum_{n>0, k>=0} T(n,k) * x^n * y^k and similarly B(x, y) with A039735. - Michael Somos, Aug 23 2015