Infinitely Based Examples

For a classDwhich admits infinite pin sequences, Theorem8.8gives us no information on

whether the basis ofC o D(here for a specified classC) is finite. However, the proof does

tell us what some of the basis elements look like. A basis elementβ of a wreath product

C o D is built around acoreof points order isomorphic to a basis element ofC. To preserve

all the points of this core when taking the D-profile of β (as required by Theorem 8.3),

every minimal block between any two points of the core must involve a basis element of

8.4 INFINITELY BASEDEXAMPLES 165

Figure 8.2:The elementβ5in the basis ofAv(25134)oAv(321).

made arbitrarily long. For example, the classAv(321)admits the increasing oscillating pin sequence encodedRU RU RU· · ·, and so we have:

Theorem 8.9. Av(25134)_oAv(321)is not finitely based.

Proof. We exhibit an infinite antichain generated by repeatedly taking up and right pins lying in the basis ofAv(25134)oAv(321). The first few elements of the antichain are

β1 = 2,5,1,3,7,6,4

β2 = 2,5,1,3,7,4,9,8,6

βk = 2,5,1,3,7,4|9,6,11,8, . . . ,2k+ 3,2k |2k+ 5,2k+ 4,2k+ 2 (k ≥3).

Here, as in [9], the|symbol is used only to clarify the structure of the permutation. See

Figure8.2for an illustration of a typical member of this antichain. We observe: (i) The set{βk |k≥1}is an antichain.

(ii) The only occurrence of321in eachβkis2k+ 5,2k+ 4,2k+ 2.

(iii) The only occurrence of25134in eachβkis2,5,1,3,·,4, and hence this forms the core.

(iv) Eachβkis neither sum nor skew decomposable.

(v) TheAv(321)-profile ofβkis2,5,1,3,7,4, . . . ,2k+ 3,2k,2k+ 4,2k+ 2(the only non-

trivial deflation occurs between2k+ 5and2k+ 4). In particular,25134≺β_kAv(321)for

It only remains to show thatβkis minimally not inAv(25134)oAv(321). Consider the effect

of removing any symbolj. Ifj= 2k+ 5,2k+ 4or2k+ 2then by (ii) this no longer involves 321 soβk−j ∈Av(321) ⊂Av(25134)oAv(321). Similarly, ifj = 2,5,1,3 or4then by (iii) βk−jno longer has a core, soβk−j∈Av(25134)⊂Av(25134)oAv(321).

For any otherj,βk−jis sum decomposable. Under theAv(321)-profile, the first (lower)

component deflates to a single point, and hence(βk−j)Av(321)∈Av(25134). Thusβk−j∈ Av(25134)oAv(321), completing the proof.

Note that in the above example, the class C = Av(25134) was specifically chosen so that the basis element25134 is not contained in the repeated pin sequence used to build the antichain, but it does lie in the class D. This ensures that the core,25134, acts as an anchor at the base of the antichain, but yet the only instance of the basis element 321 is in the upper anchor.

As a result, for any class Dwhich contains both the infinite pin sequence formed by

alternating between up and right pins, and the permutation 25134, the wreath product Av(25134)_{o D}will always contain an infinite antichain similar to the one above.

Example 8.10. (i) The classesD= Av(321,2341)and_D= Av(321,3412)both avoid the permutation321and so the antichain in the proof of Theorem8.9lies in the basis of Av(25134)_{o D}in both cases.

(ii) All of the classesD= Av(α, β)where the pair(α, β)is one of

(4321,4312),(4321,4231),(4321,4213),(4321,3412)and(4321,3214) avoid4321, and so the antichain with terms

β1 = 2,5,1,3,8,7,6,4

β2 = 2,5,1,3,7,4,10,9,8,6

βk = 2,5,1,3,7,4|9,6,11,8, . . . ,2k+ 3,2k |2k+ 6,2k+ 5,2k+ 4,2k+ 2 (k ≥3)

8.4 INFINITELY BASEDEXAMPLES 167

Figure 8.3:The elementβ5in the basis ofAv(25143)oAv(4321,4123).

(iii) The classes D = Av(4312,4231), _D = Av(4312,4213) and_D = Av(4312,3421) all avoid4312, so swapping the order of the final two points of eachβkin case (ii) gives

the required antichain.

Example 8.11. The two classesD = Av(4321,4123) andD = Av(4312,4123) both admit the pin sequence formed by repeatedly taking up and right pins, but do not contain the permutation25134, because of the basis element4123. However, the classC = Av(25143) may be used instead. In the first case, the antichain is (see Figure8.3for an illustration):

β1 = 2,5,1,4,8,7,6,3

β2 = 2,5,1,4,7,3,10,9,8,6

βk = 2,5,1,4,7,3|9,6,11,8, . . . ,2k+ 3,2k|2k+ 6,2k+ 5,2k+ 4,2k+ 2 (k ≥3).

All the examples so far have admitted the same “up-right” pin sequence, correspond- ing to variants of the increasing oscillating antichain. Another commonly found infinite pin sequence is formed by repeating the pattern left, down, right, up,1 _{and there are (to}

within symmetry) two classes of the formD = Av(α, β) with|α| = |β| = 4which admit this sequence: D = Av(3412,2413) and_D = Av(3412,2143). Each one must be handled separately.

Example 8.12. (i) D = Av(3412,2413)may be paired with _C = Av(31542) to produce

Figure 8.4:The basis elementβ3inAv(31542)oAv(3412,2413).

the antichain with terms

β1 = 8,1,6,4,9,7,5,2,3

βk = 4k+ 4,1,4k+ 2,4,4k,6, . . .2k+ 6,2k | 2k+ 4,2k+ 2,2k+ 7,2k+ 5,2k+ 3|

2k+ 9,2k+ 1, . . . ,4k+ 5,5| 2,3 (k≥2).

See Figure 8.4 for an illustration. Note that the occurrence of 3412 in anyβk is not

unique, but every occurrence requires the final two symbols2,3ofβk, and so these

points still behave in the same way as in previous examples.

(ii) D = Av(3412,2143) may be paired with_C = Av(412563) to produce the antichain with terms:

β1 = 10,1,8,4,6,9,11,7,5,2,3

βk = 4k+ 6,1,4k+ 4,4,4k+ 2,6, . . . ,2k+ 8,2k|

2k+ 6,2k+ 2,2k+ 4,2k+ 7,2k+ 9,2k+ 5,2k+ 3_| 2k+ 11,2k+ 1, . . . ,4k+ 7,5_|2,3 (k _≥2).

8.5 CONCLUDINGREMARKS ANDCONJECTURES 169