Enumerating the Derangements of an $n$-Cube via M\"obius Inversion

TL;DR

This paper develops a formula using M"obius inversion to count derangements of an n-cube by analyzing automorphisms of the face lattice that fix subalgebras, and computes the relevant M"obius function.

Contribution

It introduces a novel approach to enumerate derangements of an n-cube using lattice-theoretic M"obius inversion and computes the associated M"obius function.

Findings

Derived a formula for derangements of n-cubes.

Computed the M"obius function on the lattice of MR-subalgebras.

Provided explicit enumeration methods for automorphisms fixing subalgebras.

Abstract

In $\mathcal L$, the semilattice of faces of an $n$-cube, we count the number of automorphisms of $\mathcal L$ that fix a given subalgebra -- either pointwise or as a subalgebra. By using M\"obius inversion we get a formula for the number of derangements on the $n$-cube in terms of the M\"obius function on the lattice of MR-subalgebras. We compute this M\"obius function.