Number of cycles in the graph of 312-avoiding permutations

The graph of overlapping permutations is defined in a way analogous to the De Bruijn graph on strings of symbols. That is, for every permutation $\pi = \pi_{1} \pi_{2} ... \pi_{n+1}$ there is a directed edge from the standardization of $\pi_{1} \pi_{2} ... \pi_{n}$ to the standardization of $\pi_{2} \pi_{3} ... \pi_{n+1}$. We give a formula for the number of cycles of length $d$ in the subgraph of overlapping 312-avoiding permutations. Using this we also give a refinement of the enumeration of 312-avoiding affine permutations and point out some open problems on this graph, which so far has been little studied.