An Elementary Proof of the Explicit Formula
for the Möbius Number
of the Odd Partition Poset Kenneth M. Monks
Department of Mathematics
Front Range Community College -- Boulder County Campus
2190 Miller Drive
Longmont, CO 80537
USA
mailto:kenneth.monks@frontrange.edukenneth.monks@frontrange.edu

in

Abstract:

The Möbius number of a finite poset is a very useful combinatorial invariant of the poset that generalizes the classical number-theoretic Möbius function. The Möbius number of the poset of partitions $\Pi_n$ of a set with n elements is well-known. A related poset, the subposet consisting only of partitions that use odd part size or the maximum element $\{\{1,2,\ldots,n\}\}$, written $\Pi^{\rm odd}_n$, arises in similar combinatorial settings. In this paper, we compute the Möbius numbers of all $\Pi^{\rm odd}_n$ as follows:

$$\mu \left(\Pi_n^{\textrm{odd}}\right)= \begin{cases} (-1)^{... ...)\left((n-3)!!\right)^2, & \text{if $n$ is even.} \end{cases}$$

This result was first stated as known by Stanley and has since been proven by Sundaram and Wachs. They constructed versions of the formula above by respectively using symmetric function/representation theory and topological/simplicial complex techniques. In this paper, we provide a new proof using only elementary combinatorial techniques and the WZ algorithm.