A068182 Number of combinatorially non-equivalent "triangulations" of a compact genus n surface which have only 1 vertex (all vertices of the triangles are identified). Also the number of combinatorially distinct identifications of pairs of edges of a polygone P having 12g-6 sides leading to a compact oriented genus g surface containing the boundary of P as a 3-regular graph.

1, 9, 1726, 1349005, 2169056374, 5849686966988, 23808202021448662, 136415042681045401661, 1047212810636411989605202, 10378926166167927379808819918, 129040245485216017874985276329588

Offset

0,2

Comments

In the Krasko paper p. 18, Table 2, this sequence is designated "tautilde^(3)_(+)(g)" and has offset 1. - Michael De Vlieger, Oct 31 2021

References

R. Bacher and A. Vdovina, Counting 1-vertex triangulations of oriented surfaces, Discrete Math. 246 (2002), 13-27.

Links

Table of n, a(n) for n=0..10.

R. Bacher and A. Vdovina, Counting 1-vertex triangulations of oriented surfaces

Evgeniy Krasko, Igor Labutin, and Alexander Omelchenko, Enumeration of 3-regular one-face maps on orientable or non-orientable surface up to all symmetries, arXiv:1901.06591 [math.CO], 2019.

Formula

Reference gives a formula. Another formula can be derived by use of characters of the symmetric groups.

Example

The first term, 1, is associated to the usual construction of the torus: identify opposite sides of a square. The 1-vertex triangulation is obtained by subdividing the square into 2 triangles along a diagonal. Another point of view is to identify opposite sides of a hexagon (thus getting a torus). The 1-vertex triangulation is the dual of the boundary of the hexagon (which is a graph having 2 nodes and a triple edge between them) drawn on the torus.

Crossrefs

Cf. A348794, A348795, A348796, A348797.

Sequence in context: A322245 A251700 A246115 * A368368 A218415 A194135

Adjacent sequences: A068179 A068180 A068181 * A068183 A068184 A068185

Keyword

nonn

Author

Roland Bacher, Mar 23 2002

Status

approved