Statistics of Blocks in k-Divisible Non-Crossing Partitions

Abstract

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 $\frac{kn+1}{(k+1)^{t+1}}$. Similar results are obtained for type $B$ and type $D$ non-crossing partitions of Armstrong.