On WQO property for different quasi orderings of the set of permutations

The property of certain sets being well quasi ordered (WQO) has several useful applications in computer science - it can be used to prove the existence of efficient algorithms and also in certain cases to prove that a specific algorithm terminates. One of such sets of interest is the set of permutations. The fact that the set of permutations is not WQO has been rediscovered several times and a number of different permutation antichains have been published. However these results apply to a specific ordering relation of permutations ≤, which is not the only 'natural' option and an alternative ordering relation of permutations ⊴(more related to 'graph' instead of 'sorting' properties of permutations) is often of larger practical interest. It turns out that the known examples of antichains for the ordering ≤ can't be used directly to establish that ⊴ is not WQO. In this paper we study this alternative ordering relation of permutations ⊴ and give an example of an antichain with respect to this ordering, thus showing that ⊴ is not WQO. In general antichains for ⊴ cannot be directly constructed from antichains for ≤, however the opposite is the case - any antichain for ⊴ allows to construct an antichain for ≤.