There is a formula in terms of conjugacy classes of transitive pairs and triples of permutations of degree n (A057005 and A057006).

REFERENCES

N. G. Markley and W. N. Anderson,jun, An application of Moebius inversion to a problem in topological dynamics. Bull. Amer. Math. Soc. 79 (1973), 1027-1029.

V. A. Liskovets, On a combinatorial problem of the theory of topological transformations of the circle. Vesci AN BSSR (ser. fiz.-mat.), No 4 (1976), 29-34 (in Russian), Math. Rev. 58 #7575.