  The author has already proven that the space $\Delta(\Pi_n)/G$ is homotopy
equivalent to a wedge of spheres of dimension $n-3$ for all natural numbers
$n\geq 3$ and all subgroups $G\subset S_1\times S_{n-1}$. We construct an
$S_1\times S_{n-1}$-equivariant acyclic matching on $\Delta(\Pi_n)$ together
with a description of its critical simplices. This is also a more elementary
approach to determining the number of spheres. We also develop new methods for
Equivariant Discrete Morse Theory by adapting the Patchwork Theorem and poset
maps with small fibers from Discrete Morse Theory.
