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
Andrew Howroyd, Table of n, a(n) for n = 0..300
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
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