Moving faces to other places: Facet derangements

TL;DR

This paper explores face derangements of n-dimensional hypercubes, classifying them as odd or even based on isometry type, and highlights their connections to geometry, algebra, and combinatorics.

Contribution

It introduces a novel generalization of derangements to hypercube faces, linking geometric, algebraic, and combinatorial perspectives.

Findings

Classification of face derangements into odd and even types

Connection between derangements and hypercube isometries

Illustrations demonstrating geometric and algebraic properties

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.