login
A292206
Number of unrooted unlabeled connected four-regular maps on a compact closed oriented surface with n vertices (and thus 2*n edges).
3
1, 2, 7, 36, 365, 5250, 103801, 2492164, 70304018, 2265110191, 82013270998, 3295691020635, 145553281837454, 7008046130978980, 365354356543414133, 20504381826687810441, 1232562762503125498772, 79012106044626365750974, 5380476164948914549410335, 387882486153123498708054879
OFFSET
0,2
COMMENTS
Equivalently, the number of unrooted quadrangulations of oriented surfaces with n quadrangles (and thus 2*n edges).
Equivalently, the number of pairs (alpha,sigma) of permutations on a set of size 4*n up to simultaneous conjugacy such that alpha (resp. sigma) has only cycles of length 2 (resp. 4) and the subgroup generated by them acts transitively.
There is no recurrence relation known for this sequence.
LINKS
Laura Ciobanu and Alexander Kolpakov, Free subgroups of free products and combinatorial hypermaps, Discrete Mathematics, 342 (2019), 1415-1433; arXiv:1708.03842 [math.CO], 2017-2019.
FORMULA
Inverse Euler transform of A268556. - Andrew Howroyd, Jan 29 2025
EXAMPLE
For n = 1, a(n) = 2:
1) the figure-eight map on a sphere (1 vertex, which has degree 4, and 2 edges) <-> its dual map, which is the quadrangulation of a sphere created by a 2-edge path (it bounds 1 region, which has 4 boundary segments, even though they are formed by only 2 different edges) <-> the conjugacy class of the pair of permutations ((12)(34), (1234));
2) the map on a torus consisting of two non-homotopic nontrivial loops (1 vertex, which has degree 4, and 2 edges) <-> its dual map, which is the same map again (it bounds 1 region, which has 4 boundary segments, even though they are formed by only 2 different edges) <-> the conjugacy class of the pair of permutations ((13)(24), (1234)).
CROSSREFS
Column 4 of A380626.
Unrooted version of A292186.
Cf. A268556.
Sequence in context: A007474 A002724 A348106 * A203900 A209251 A366670
KEYWORD
nonn
AUTHOR
Sasha Kolpakov, Sep 11 2017
EXTENSIONS
Edited by Andrey Zabolotskiy, Jan 17 2025
a(0)=1 prepended and a(18) onwards from Andrew Howroyd, Jan 29 2025
STATUS
approved