A112945
Number of unrooted regular odd-valent planar maps with 4 vertices; maps are considered up to orientation-preserving homeomorphisms and the vertices are of valency 2n+1.
Original entry on oeis.org
0, 6, 468, 80600, 16016560, 3360790440, 728936019504, 161858688461184, 36580777518027600, 8382066029146609800, 1941971956789550319920, 454006489072843947528288, 106944132919124515725427808
Offset: 0
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a[n_] := (n/6) Binomial[2n, n]^4 + (n/2) Binomial[2n, n]^2 + (2/3) Boole[ Divisible[2n+1, 3]] Binomial[2 Floor[n/3], Floor[n/3]] Binomial[2n, n];
Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Aug 29 2019 *)
A113181
Number of unrooted two-vertex (or, dually, two-face) regular planar maps of even valency 2n considered up to orientation-preserving homeomorphism.
Original entry on oeis.org
1, 3, 14, 95, 859, 9130, 106039, 1297295, 16428300, 213388961, 2827645453, 38086408002, 520062618300, 7184570776213, 100256059855188, 1411319038583375, 20021022607979629, 285965560309310708, 4109498933510809561, 59380204746202961953, 862266486434574492404
Offset: 1
There exist 3 planar maps with two 4-valent vertices: a map with four parallel edges and two different maps with two parallel edges and one loop in each vertex. Therefore a(2)=3.
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a[n_] := Binomial[2n, n]/4 + (1/(4n)) Sum[EulerPhi[k] Binomial[2n/k - 1, Floor[n/k]]^2, {k, Divisors[2n]}];
Array[a, 21] (* Jean-François Alcover, Jul 24 2018 *)
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a(n) = binomial(2*n, n)/4 + sumdiv(2*n, k, eulerphi(k)* binomial(2*n/k-1, (n\k))^2)/(4*n); \\ Michel Marcus, Oct 14 2015
A113182
Number of unrooted two-vertex (or, dually, two-face) regular planar maps of valency n considered up to orientation-preserving homeomorphism.
Original entry on oeis.org
1, 1, 2, 3, 7, 14, 39, 95, 308, 859, 3013, 9130, 33300, 106039, 394340, 1297295, 4878109, 16428300, 62232321, 213388961, 812825244, 2827645453, 10818489817, 38086408002, 146250545528, 520062618300, 2003199281223, 7184570776213
Offset: 1
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a[n_] := If[OddQ[n], (1/(2n))(Sum[EulerPhi[d] Binomial[2 Floor[(n-1)/(2d)], Floor[(n-1)/(2d)]]^2, {d, Divisors[n]}] + n Binomial[n-1, (n-1)/2]), (1/4)((2 Sum[EulerPhi[d] Binomial[n/d-1, Floor[n/(2d)]]^2, {d, Divisors[ n]}])/n + Binomial[n, n/2])];
Array[a, 28] (* Jean-François Alcover, Aug 30 2019 *)
A112948
Number of unrooted 3-regular planar maps with 2n vertices, up to orientation-preserving isomorphisms.
Original entry on oeis.org
2, 6, 26, 191, 1904, 22078, 282388, 3848001, 54953996, 814302292
Offset: 1
There exist 2 planar maps with two 3-valent vertices: a map with three parallel edges and a map with one loop in each vertex and a link connecting the vertices. Therefore a(1)=2.
- Z. C. Gao, V. A. Liskovets and N. C. Wormald, Enumeration of unrooted odd-valent regular planar maps, Preprint, 2005.
- Mark van Hoeij, Vijay Jung Kunwar, Classifying (near)-Belyi maps with Five Exceptional Points, arXiv preprint arXiv:1604.08158, 2016. Also in Indagationes Mathematicae (2019) Vol. 30, No. 1, 136-156.
- Riccardo Murri, Fatgraph algorithms and the homology of the Kontsevich complex, arXiv preprint arXiv:1202.1820, 2012.
3-regular maps on the torus:
A292408.
Showing 1-4 of 4 results.
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