Chain enumeration of $k$-divisible noncrossing partitions of classical types

TL;DR

This paper provides combinatorial proofs for formulas counting multichains in $k$-divisible noncrossing partitions of classical types and confirms Armstrong's conjecture on the zeta polynomial for type A partitions with rotational symmetry.

Contribution

It offers new combinatorial proofs for existing formulas and proves Armstrong's conjecture on the zeta polynomial for type A partitions with symmetry.

Findings

Confirmed formulas for multichains in $k$-divisible noncrossing partitions

Proved Armstrong's conjecture on the zeta polynomial for symmetric type A partitions

Enhanced understanding of the combinatorial structure of noncrossing partitions

Abstract

We give combinatorial proofs of the formulas for the number of multichains in the $k$-divisible noncrossing partitions of classical types with certain conditions on the rank and the block size due to Krattenthaler and M{\"u}ller. We also prove Armstrong's conjecture on the zeta polynomial of the poset of $k$-divisible noncrossing partitions of type $A$ invariant under a $180^\circ$ rotation in the cyclic representation.