Enumerative Properties of NC^B(p,q)

TL;DR

This paper investigates the enumerative properties of annular non-crossing partitions of type B, providing explicit formulas for their rank generating functions, zeta polynomials, and Moebius functions, and extends results to multiannular cases.

Contribution

It offers new explicit formulas and an alternative approach for the case q=1, and generalizes to multiannular non-crossing partitions of type B.

Findings

Derived the rank generating function for NC^B(p,q)

Computed the zeta polynomial and Moebius function for these posets

Reduced multiannular cases to simpler non-crossing partition cases

Abstract

We determine the rank generating function, the zeta polynomial and the Moebius function for the poset NC^B(p,q) of annular non-crossing partitions of type B, where p and q are two positive integers. We give an alternative treatment of some of these results in the case q=1, for which this poset is a lattice. We also consider the general case of multiannular non-crossing partitions of type B, and prove that this reduces to the cases of non-crossing partitions of type B in the annulus and in the disc.