The M？bius function of partitions with restricted block size

We study filters in the partition lattice formed by restricting to partitions by type. The M\"obius function is determined in terms of the easier-to-compute descent set statistics on permutations and the M\"obius function of filters in the lattice of integer compositions. When the underlying integer partition is a knapsack partition, the M\"obius function on integer compositions is determined by a topological argument. In this proof the permutahedron makes a cameo appearance.