  In their study of cyclic pattern containment, Domagalski et al. conjecture
differential equations for the generating functions of circular permutations
avoiding consecutive patterns of length 3. In this note, we prove and
significantly generalize these conjectures. We show that, for every consecutive
pattern $\sigma$ beginning with 1, the bivariate generating function counting
occurrences of $\sigma$ in circular permutations can be obtained from the
generating function counting occurrences of $\sigma$ in (linear) permutations.
This includes all the patterns for which the latter generating function is
known.
