An Elementary Proof of the Explicit Formula for the Möbius Number of the Odd Partition Poset

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:

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

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.

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

Kenneth M. Monks
Department of Mathematics
Front Range Community College -- Boulder County Campus
2190 Miller Drive
Longmont, CO 80537
USA