The M\"obius function of partitions with restricted block size

TL;DR

This paper investigates the M"obius function in the lattice of partitions with restricted block sizes, connecting it to permutation statistics and lattice theory, with special cases involving knapsack partitions and topological methods.

Contribution

It introduces a novel approach to compute the M"obius function for partitions with restrictions using permutation descent sets and topological arguments involving the permutahedron.

Findings

M"obius function expressed via descent set statistics

Connection established with lattice of integer compositions

Topological proof involving permutahedron for knapsack partitions

Abstract

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.