Unshuffling permutations: Trivial bijections and compositions

Abstract

Given permutations π, σ1 and σ2, the permutation π (viewed as a string) is said to be a shuffle of σ1 and σ2, in symbols (formula presented), if π can be formed by interleaving the letters of two strings p1 and p2 that are order-isomorphic to σ1 and σ2, respectively. Given a permutation π ∈ S2n and a bijective mapping f:Sn→f:Sn, the f-Unshuffle-Permutation problem is to decide whether there exists a permutation σ∈Sn such that (formula presented). We consider here this problem for the following bijective mappings: inversion, reverse, complementation, and all their possible compositions. In particular, we present combinatorial results about the permutations accepted by this problem. As main results, we obtain that this problem is NP -complete when f is the reverse, the complementation, or the composition of the reverse with the complementation.