  In this paper, we focus on the enumeration of permutations by number of
cyclic occurrence of peaks and valleys. We find several recurrence relations
involving the number of permutations with a prescribed number of cyclic peaks,
cyclic valleys, fixed points and cycles. Several associated permutation
statistics and the corresponding generating functions are also studied. In
particular, we establish a connection between cyclic valleys and Pell numbers
as well as cyclic peaks and alternating runs.
