Unshuffling Permutations

TL;DR

This paper explores the algebraic properties and computational complexity of recognizing square permutations, introducing a novel unshuffling operator and establishing NP-completeness for the recognition problem.

Contribution

It introduces an unshuffling operator for permutations, provides algebraic and combinatorial insights, and proves the NP-completeness of recognizing square permutations.

Findings

Algebraic and combinatorial properties of the shuffle product

Bijection between square (213,231)-avoiding permutations and square binary words

Recognition of square permutations is NP-complete

Abstract

A permutation is said to be a square if it can be obtained by shuffling two order-isomorphic patterns. The definition is intended to be the natural counterpart to the ordinary shuffle of words and languages. In this paper, we tackle the problem of recognizing square permutations from both the point of view of algebra and algorithms. On the one hand, we present some algebraic and combinatorial properties of the shuffle product of permutations. We follow an unusual line consisting in defining the shuffle of permutations by means of an unshuffling operator, known as a coproduct. This strategy allows to obtain easy proofs for algebraic and combinatorial properties of our shuffle product. We besides exhibit a bijection between square $(213,231)$-avoiding permutations and square binary words. On the other hand, by using a pattern avoidance criterion on oriented perfect matchings, we prove that recognizing square permutations is $\mathbf{NP}$-complete.