Statistics of blocks in k-divisible non-crossing partitions

We derive a formula for the expected number of blocks of a given size from a non-crossing partition chosen uniformly at random. Moreover, we refine this result subject to the restriction of having a number of blocks given. Furthermore, we generalize to k-divisible partitions. In particular, we find that, asymptotically, the expected number of blocks of size t of a k-divisible non-crossing partition of nk elements chosen uniformly at random is (kn+1)/(k+1)^(t+1). Similar results are obtained for type B and type D k-divisible non-crossing partitions of Armstrong.