Permutation patterns in genome rearrangement problems

TL;DR

This paper explores permutation patterns within genome rearrangement models, specifically block transpositions, providing characterizations and properties of permutations close to the identity for various distances.

Contribution

It introduces a novel characterization of permutations near the identity in genome rearrangement models using generating permutations and analyzes their basis properties.

Findings

Characterization of permutations with distance ≤ k from the identity.

Description of properties of the basis of permutation classes.

Method to compute the basis for small values of k.

Abstract

In the context of the genome rearrangement problem, we analyze two well known models, namely the block transposition and the prefix block transposition models, by exploiting the connection with the notion of permutation pattern. More specifically, for any $k$, we provide a characterization of the set of permutations having distance $\leq k$ from the identity (which is known to be a permutation class) in terms of what we call generating permutations and we describe some properties of its basis, which allow to compute such a basis for small values of $k$.