Suppose we have a Coxeter group whose diagram is given by a simplex. In other words, $G = \langle g_1,...g_k | (g_i)^2 = e, (g_ig_j)^3 = e \rangle$. How many words of length $N$ simplify to the identity? What is the recursion/generating function? The case $k=2$ is easy, because the group is finite; the corresponding generating function is $E(x) = \frac{1}{3} [2/(1-x^2) + 1/(1-4x^2)]$. I expect $k=3$ is likewise readily doable. Is there a general solution? What if we change the relations to $(g_ig_j)^m = e \,\,\forall i,j$?
Keep in mind the word problem is solvable for Coxeter groups.
More generally, if I give a group element in this group whose shortest word is $g_{i_1}...g_{i_p}$, how many words of length $N$ are equivalent to it?