Buan, Reiten and Thomas (implicitly) defined positive $m$-divisible non-crossing partitions, by setting up a bijection between the facets of the generalized cluster complex of Fomin and Reading and m-divisible non-crossing partitions, positive clusters corresponding to positive $m$-divisible non-crossing partitions. We embark on a finer enumerative study of these combinatorial objects associated with finite reflection groups. In particular, we define a cyclic action on them, which - together with the "obvious" q-analogue of positive Fu{\ss}--Catalan numbers - satisfies the cyclic sieving phenomenon of Reiner, Stanton and White. Our proof is a - lengthy - case-by-case verification. Crucial in the proof in the classical types are a combinatorial realisation of the positive m-divisible non-crossing as certain classical m-divisible non-crossing partitions on the one hand and the combinatorial description of the action as some kind of pseudo-rotation on the other hand.