I PLANTEAMIENTO TEORICO

While every basis forms an antichain, be it finite or infinite, we may also be interested in whether a class contains infinite antichains. A partial order is said to be apartial well or- derif it contains neither an infinite properly decreasing sequence nor an infinite antichain. In the case of permutation classes this first condition is always true (by the existence of a smallest element), and so a permutation class ispartially well orderedif it contains no infi- nite antichain. For example, Knuth [76] shows that the set of stack sortable permutations, Av(231)is partially well ordered.

The decidability problem of whether a given permutation class is partially well ordered remains open:

Question 5.15. Is it possible to decide if a permutation class given by a finite basis is partially well ordered?6

Indeed there has been no recent major progress on the general problem. Alongside a variety of specific examples, Atkinson, Murphy and Ruˇskuc [9] showed that Av(β) is partially well ordered if and only ifβ ∈ {1,12,21,132,213,231,312}.

Showing that a class is not partially well ordered is simply a case of spotting an an- tichain inside it. For example, the class Av(321) contains the increasing oscillating an- tichain presented above. A non-partially well ordered class may contain many infinite antichains, but among them there must be at least one fundamental antichain.

Proposition 5.16(Gustedt [66]). Every non-partially well ordered permutation class contains an infinite fundamental antichain.

Proof. With an eye toward applying Zorn’s lemma, take an infinite descending chainA1

A2 _{· · ·} of infinite antichains and define

A∞={α:αis an element of all but finitely manyAis}.

5.3 ANTICHAINS, PARTIALWELLORDER ANDATOMICITY 99

First observe that A∞ is an antichain, and that A∞ Ai for all i. We claim that it is

also infinite. Suppose to the contrary that A∞ is finite. Thus _A∞ is a subset of all but finitely many of theAis; without loss let us assume that it is contained in all theAis. Now

chooseα1 ∈ A1\A∞. For each i ≥ 2, because Ai Ai−1, we may chooseαi ∈ Ai such

that αi ≤ αi−1. This gives a descending chain α1 ≥ α2 ≥ . . ., so because permutation classes have no infinite strictly descending chains, there is some α∞ and integerI such thatαi =α∞for alli ≥I. However, this implies thatα1 ≥ αI = α∞ ∈ A∞ ⊆A1, which requires (becauseA1 is an antichain)α1 = α∞, a contradiction to our choice ofα1. Thus Zorn’s Lemma shows that the set of infinite antichains in a non-partially well ordered class has a minimal element under, as desired.

Note that if A is a fundamental antichain then itsstrict closure, _{π : π < α _∈ A_}, is

partially well ordered.

On the other hand, showing that a class is partially well ordered is a considerably harder task. The primary tool here is a result of Higman [67], which we now state. We say that(A, M)is anabstract algebraifAis a set of elements andM a set of operations, for

which eachµ ∈ M is ak-ary operation,µ : Ak → A, for some positive integerk. Denote

the set ofk-ary operations byMk, and suppose thatMkis empty for everyk > nfor some n. (Note that we will allow 0-ary operations.) The abstract algebra (A, M) is said to be

minimalif no subsetBofAallows(B, M)to be an abstract algebra.

A partial order≤A on the set of elements A is a divisibility order on(A, M) if every

operationµ_∈Mk,k= 0,1, . . . , n, satisfies,

• a≤Abimpliesµ(x, a,y) ≤Aµ(x, b,y),

• a_≤Aµ(x, a,y),

where xandy are arbitrary sequences comprising elements of A whose lengths sum to k ₋1. Furthermore, given partial orders_≤Mk onMk, k = 0,1, . . . , n, we say that ≤A is compatiblewith these partial orders if, forλ, µ∈Mk,

Theorem 5.17(Higman [67]). Suppose that(A, M)is a minimal abstract algebra for which, for

somen, the setMkofk-ary operations inM is partially well ordered for eachk = 0,1, . . . , nand empty fork > n. Then(A, M)is partially well ordered under any divisibility ordering compatible

with the orders ofMk.

Higman’s Theorem is applied to prove that a given permutation class is partially well ordered by showing how we may “build” the class from a smaller (very possibly finite) set.

Example 5.18. By our definition in Example5.4, the classAv(2413,3142)of separable per- mutations is precisely the strong completion of the class {1_}, i.e. the class formed from the permutation 1 using the binary operations_⊕ and . Higman’s Theorem may now immediately be applied to show thatAv(2413,3142)is partially well ordered.

A permutation classC isstrongly finitely based if it is finitely based and every closed

subset ofC is also finitely based.7 Recalling that the basis of a class is an antichain, this

definition immediately returns us to partial well order, and indeed we have a variety of equivalent conditions. A formal proof is provided by Atkinson, Murphy and Ruˇskuc [9]. Proposition 5.19. LetCbe a permutation class. Then the following are equivalent:

(1) Cis strongly finitely based.

(2) Chas at most countably many closed subsets.

(3) Ccontains no infinite antichain.

(4) The subclasses ofCsatisfy the descending chain condition.

Partial well order also plays a rˆole in some enumeration attempts. Klazar [75] shows that the smallest growth rate which admits uncountably many closed permutation classes lies between2and2.33529. . .. This growth rate is determined by the smallest growth rate

that a non-partially well ordered class can have – by Proposition 5.19, such a class will

5.3 ANTICHAINS, PARTIALWELLORDER ANDATOMICITY 101

have uncountably many closed subsets, each of which cannot have a growth rate larger than the parent class. The lower bound arises by showing all classes with growth rate under 2 contain only finitely long alternations and oscillations, and these classes – via Higman – are partially well ordered. The upper bound arises by considering the class Av(321,4123,3412,23451), and noting that it contains the increasing oscillating antichain (hence is not partially well ordered). This class has rational generating function

f(x) = x

5_+x4_+x3_+x2_+x

1−x−2x2_−2x3_−x4_−x5

and the growth rate2.33529. . . arises as the reciprocal of the smallest real root of the de-

nominator (in fact, it is the only real root). Klazar mentions that Vatter and Murphy [pri- vate communication] can improve the upper bound to2.20556. . .. The class which satis-

fies this is formed by appending the basis elements134526, 134625,314526and314625to Av(321,4123,3412,23451), and its growth rate is the dominant root ofx3−2x2−1.

More recently, Vatter [117] proved that the bound is precisely2.20556. . . by computing

the growth rates of all partially well ordered classes, a task relying on Proposition5.22. He also makes the following conjecture:

Conjecture 5.20. Every growth rate of permutation classes is also the growth rate of a partially well ordered permutation class.