  We show an instance of the cyclic sieving phenomenon on annular noncrossing
permutations with given cycle types. We define annular $q$-Kreweras numbers,
annular $q$-Narayana numbers, and annular $q$-Catalan number, all of which are
polynomials in $q$. We then show that these polynomials exhibit the cyclic
sieving phenomenon on annular noncrossing permutations. We also show that a sum
of annular $q$-Kreweras numbers becomes an annular $q$-Narayana number and a
sum of $q$-Narayana numbers becomes an annular $q$-Catalan number.
