A292206 Number of unrooted unlabeled connected four-regular maps on a compact closed oriented surface with n vertices (and thus 2*n edges).
1, 2, 7, 36, 365, 5250, 103801, 2492164, 70304018, 2265110191, 82013270998, 3295691020635, 145553281837454, 7008046130978980, 365354356543414133, 20504381826687810441, 1232562762503125498772, 79012106044626365750974, 5380476164948914549410335, 387882486153123498708054879
Offset: 0
Keywords
Examples
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)).
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
Extensions
Edited by Andrey Zabolotskiy, Jan 17 2025
a(0)=1 prepended and a(18) onwards from Andrew Howroyd, Jan 29 2025
Comments