  The circular peak set of a permutation $\sigma$ is the set $\{\sigma(i)\mid
\sigma(i-1)<\sigma(i)>\sigma(i+1)\}$. In this paper, we focus on the
enumeration problems for permutations by circular peak sets. Let $cp_n(S)$
denote the number of the permutations of order $n$ which have the circular peak
set $S$. For the case with $|S|=0,1,2$, we derive the explicit formulas for
$cp_n(S)$. We also obtain some recurrence relations for the sequence $cp_n(S)$
and give the formula for $cp_n(S)$ in the general case.
