Statistics of blocks in k-divisible non-crossing partitions

TL;DR

This paper derives formulas for the expected number of blocks of a given size in k-divisible non-crossing partitions, providing asymptotic results and generalizations to types B and D.

Contribution

It introduces new formulas and asymptotic results for block size distributions in k-divisible non-crossing partitions, extending previous work to more general types.

Findings

Expected number of blocks of size t asymptotically (kn+1)/(k+1)^(t+1)

Refined formulas for fixed number of blocks

Generalizations to type B and D 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 (kn+1)/(k+1)^(t+1). Similar results are obtained for type B and type D k-divisible non-crossing partitions of Armstrong.