The complex of injective words of permutations which are not derangements is contractible

TL;DR

This paper proves that the complex formed by injective words from permutations that are not derangements is contractible, providing a conceptual explanation for the homotopy type of the complex generated by all permutations.

Contribution

It introduces a new contractibility result for a specific complex of injective words, extending understanding of the homotopy types related to permutation groups.

Findings

The complex of injective words from non-derangements is contractible.

Provides a conceptual proof for the homotopy equivalence involving derangements.

Connects the structure of injective words with topological properties of permutation complexes.

Abstract

Let $D_n \subseteq \Sigma_n$ be the set of derangements in the symmetric group. We prove that the complex of injective words generated by $\Sigma_n \setminus D_n$ is contractible. This gives a conceptual explanation to the well known fact that the complex of injective words generated by $\Sigma_n$ is homotopy equivalent to the wedge sum $\underset{|D_n|}{\bigvee} S^{n-1}$.