Posets of annular non-crossing partitions of types B and D

TL;DR

This paper explores the structure of annular non-crossing partitions of types B and D, establishing poset isomorphisms and lattice properties, thereby extending combinatorial understanding of these algebraic objects.

Contribution

It introduces and proves the poset isomorphism between annular non-crossing permutations and partitions of types B and D, including lattice structures for specific cases.

Findings

Poset isomorphism between permutations and partitions of types B and D.

The poset $ cb (p,1)$ is a lattice under reverse refinement.

The poset $ cd (p,1)$ coincides with a known lattice from prior work.

Abstract

We study the set $\sncb (p,q)$ of annular non-crossing permutations of type B, and we introduce a corresponding set $\ncb (p,q)$ of annular non-crossing partitions of type B, where $p$ and $q$ are two positive integers. We prove that the natural bijection between $\sncb (p,q)$ and $\ncb (p,q)$ is a poset isomorphism, where the partial order on $\sncb (p,q)$ is induced from the hyperoctahedral group $B_{p+q}$, while $\ncb (p,q)$ is partially ordered by reverse refinement. In the case when $q=1$, we prove that $\ncb (p,1)$ is a lattice with respect to reverse refinement order. We point out that an analogous development can be pursued in type D, where one gets a canonical isomorphism between $\sncd (p,q)$ and $\ncd (p,q)$. For $q=1$, the poset $\ncd (p,1)$ coincides with a poset ``$NC^{(D)} (p+1)$'' constructed in a paper by Athanasiadis and Reiner in 2004, and is a lattice by the results of that paper.