This is a redo of my earlier question.

I'm trying again with more precision. I ignored one of the dicta in the "how to ask" page.

For a fixed $n$, what is known (references preferred) about the maximum number of edge 3-colorings (yes, I known the title says (vertex) 4 colorings, but it reduces to the same question) among the following graphs: Planar, simplicial (two vertices have at most one edge between them and there are no 1-gons), triangulations with $n$ vertices so that every loop of three edges bounds a triangle. In the count, it would be best to mod out by the action of $S_3$ on the colors.

I know this number is on the order of $2^{n-4}$. I am looking for references on work on getting exact values. If f(n) is the number, then what is known about the integer sequence f(n).

The considered triangulations have Hamiltonian circuits [Whitney, Ann. of Math. 32 (1931), 378-390]. The restriction that two vertices cannot have two edges between them is not only a hypothesis of Whitney's theorem, but it also rules out a reduction of a given triangulation to a "gluing" of two triangulations giving the number of colorings as a product of the numbers for the two "glued" triangulations.