Simple permutations: Decidability and unavoidable substructures

Abstract

We prove that it is decidable if a finitely based permutation class contains infinitely many simple permutations, and establish an unavoidable substructure result for simple permutations: every sufficiently long simple permutation contains an alternation or oscillation of length k.

On Permutation Pattern Classes with Two Restrictions Only

“…They are presented as plots of points in the (x, y)-plane: p 1 p 2 · · · p n is represented as the set of n points (i, p i ). These diagrams ( Figure 1, Figure 2, Figure 3, and Figure 4) make it easy to check (as also verified in [6]) that the permutations are indeed simple and that they avoid the various permutations of small length stated. …”

“…Thus it is of great interest to know when a pattern class has just a finite number of simple permutations. A decision procedure to decide this question (given a finitely based class) has recently been given [6]; in principle it gives a bound on the length of the longest simple permutation in the class (when this exists) but the bound is almost certainly very excessive.…”

Well quasi-order in combinatorics: embeddings and homomorphisms

“…A common feature of all the antichains presented there is that the main body of each permutation consists of what was later termed a pin sequence [10], with an irregularity at the beginning and the end. This, and some subsequent results on pin sequences [11] give some hope that a (constructive) characterisation of finitely based wqo classes may be possible, but so far this has been elusive. …”

“…All the results in this paper are constructive, allowing, at least in principle, generating functions to be computed from the set of simple permutations and the basis. To complement this, Brignall, Ruškuc and Vatter [11] prove that it is decidable whether the set of simple permutations in a pattern class given by a finite basis is finite.…”

“…This is basically due to the non-constructive nature of appeals to Higman's Theorem in the argument. In a rare example of full grid classes behaving better than their geometric subclasses, Huczynska and Vatter [11] give a very nice constructive criterion for a finitely based class to be contained in a grid class. A different example of encoding permutations in a class by a regular language, in order to prove rationality of the generating function, can be found in [3].…”