Moving faces to other places: Facet derangements

Abstract

Derangements are a popular topic in combinatorics classes. We study a generalization to face derangements of the n-dimensional hypercube. These derangements can be classified as odd or even, depending on whether the underlying isometry is direct or indirect, providing a link to abstract algebra. We emphasize the interplay between the geometry, algebra, and combinatorics of these sequences, with lots of pretty pictures.