A Gray Code for the Shelling Types of the Boundary of a Hypercube

TL;DR

This paper introduces a Gray code for encoding shelling types of a hypercube boundary using indecomposable permutations, providing an efficient way to traverse all shellings with minimal changes.

Contribution

It constructs a transposition Gray code for a class of permutations representing hypercube shellings, extending previous results to a signed variant.

Findings

Constructed a Gray code for shelling types of hypercube boundaries.

Established a bijection between permutations and shelling classes.

Extended King's result to a signed permutation Gray code.

Abstract

We consider two shellings of the boundary of the hypercube equivalent if one can be transformed into the other by an isometry of the cube. We observe that a class of indecomposable permutations, bijectively equivalent to standard double occurrence words, may be used to encode one representative from each equivalence class of the shellings of the boundary of the hypercube. These permutations thus encode the shelling types of the boundary of the hypercube. We construct an adjacent transposition Gray code for this class of permutations. Our result is a signed variant of King's result showing that there is a transposition Gray code for indecomposable permutations.