3,3

Definition of c-net: a 3-connected planar map, rooted by a directed edge on the outer face.

Manuel Bodirsky, Clemens Groepl, Daniel Johannsen and Mihyun Kang, A Direct Decomposition of 3-connected Planar Graphs,conference paper (FPSAC05)

R. C. Mullin and P. J. Schellenberg, The enumeration of c-nets via quadrangulations, Journal of Combinatorial Theory 4 (1968), 259-276

M. Bodirsky, C. Groepl, D. Johannsen and M. Kang, A Direct Decomposition of 3-connected Planar Graphs.

c(0)=1, c(1) = 1, c(2) = 7, c(3) = 73, c(4) = 879, c(5) = 11713, c(6) = 167423, c(7) = 2519937, c(n) = ( (-189665280 + 134270976 n - 31309824 n^2 + 2408448 n^3) c(n-7) + (-479162880 + 376680448 n - 98932224 n^2 + 8692736 n^3) c(n-6) + (-446660160 + 384601888 n - 112131264 n^2 + 11026784 n^3) c(n-5) + (-183645792 + 168826836 n - 52598160 n^2 + 5361276 n^3) c(n-4) + (-25324080 + 24563948 n - 6853668 n^2 + 418816 n^3) c(n-3) + (1156086 - 2064937 n + 1206966 n^2 - 180467 n^3) c(n-2) + (-3192 + 4842 n - 29796 n^2 + 18930 n^3) c(n-1) ) / (126 + 693 n + 1134 n^2 + 567 n^3). Generating function C(t)=sum_(n>=0){c(n-3)t^n} implicitly given by: 0 = -1 + C(t) + 36 t - 43 C(t) t + 6 C(t)^2 t + 131 t^2 - 337 C(t) t^2 + 218 C(t)^2 t^2 + 12 C(t)^3 t^2 + 350 t^3 - 1021 C(t) t^3 + 894 C(t)^2 t^3 - 228 C(t)^3 t^3 + 8 C(t)^4 t^3 + 540 t^4 - 1828 C(t) t^4 + 2190 C(t)^2 t^4 - 988 C(t)^3 t^4 + 72 C(t)^4 t^4 + 616 t^5 - 2404 C(t) t^5 + 3284 C(t)^2 t^5 - 1756 C(t)^3 t^5 + 264 C(t)^4 t^ 5 + 536 t^6 - 2128 C(t) t^6 + 3120 C(t)^2 t^6 - 2032 C(t)^3 t^6 + 504 C(t)^4 t^6 + 304 t^7 - 1344 C(t) t^7 + 2304 C(t)^2 t^7 - 1792 C(t)^3 t^7 + 528 C(t)^4 t^7 + 160 t^8 - 768 C(t) t^8 + 1344 C(t)^2 t^8 - 1024 C(t)^3 t^8 + 288 C(t)^4 t^8 + 64 t^9 - 256 C(t) t^9 + 384 C(t)^2 t^9 - 256 C(t)^3 t^9 + 64 C(t)^4 t^9. Explicit generating function can be obtained by mathematica.

c(0)=c(1)=1 because the only c-nets on 3 respectively 4 vertices are the complete graphs.

c[0] = 1; c[1] = 1; c[2] = 7; c[3] = 73; c[4] = 879; c[5] = 11713; c[6] = 167423; c[7] = 2519937; c[n_] := ( (-189665280 + 134270976 n - 31309824 n^2 + 2408448 n^3) c[n - 7] + (-479162880 + 376680448 n - 98932224 n^2 + 8692736 n^3) c[n - 6] + (-446660160 + 384601888 n - 112131264 n^2 + 11026784 n^3) c[n - 5] + (-183645792 + 168826836 n - 52598160 n^2 + 5361276 n^3) c[n - 4] + (-25324080 + 24563948 n - 6853668 n^2 + 418816 n^3) c[n - 3] + (1156086 - 2064937 n + 1206966 n^2 - 180467 n^3) c[n - 2] + (-3192 + 4842 n - 29796 n^2 + 18930 n^3) c[n - 1] ) / (126 + 693 n + 1134 n^2 + 567 n^3);

Cf. A001506, A001507, A001508.

Sequence in context: A092444 A099141 A084768 this_sequence A114429 A124547 A084363

Adjacent sequences: A106648 A106649 A106650 this_sequence A106652 A106653 A106654

easy,nonn

Daniel Johannsen (johannse(AT)informatik.hu-berlin.de), May 12 2005