M\"obius numbers of some modified generalized noncrossing partitions

TL;DR

This paper computes the M"obius number for a specific subset of generalized noncrossing partitions associated with Coxeter groups, confirming a conjecture in the literature.

Contribution

It provides an explicit M"obius number for a modified class of noncrossing partitions, affirming a previously conjectured value.

Findings

Confirmed the conjecture 3.7.9 in Armstrong's paper.

Derived the M"obius number for the modified noncrossing partitions.

Enhanced understanding of the combinatorial structure of Coxeter groups.

Abstract

In this paper we will give a M\"obius number of $NC^{k}(W) \setminus \bf{mins} \cup \{\hat{0} \}$ for a Coxeter group $W$ which contains an affirmative answer for the conjecture 3.7.9 in Armstrong's paper [ Generalized noncrossing partitions and combinatorics of Coxeter groups. arXiv:math/0611106].