1.5 Inflations and the Substitution Decomposition
1.5.1 The Permutation Case
Restricting our attention to the permutation case, the substitution decomposition is some- what easier to describe. Given a permutationσ of lengthmand nonempty permutations α1, . . . , αm, theinflation of σ by α1, . . . , αm – denotedσ[α1, . . . , αm] – is the permutation
obtained by replacing each entry σ(i) by an interval that is order isomorphic to αi. For
Figure 1.5:The plot of479832156, an inflation of2413.
any expression ofπas an inflationπ =σ[π1, π2, . . . , πm], and we will callσaskeletonofπ.
Theorem1.5then specialises to become:
Proposition 1.7(Albert and Atkinson [2]). Every permutation may be written as the inflation of a unique simple permutation. Moreover, ifπcan be written asσ[α1, . . . , αm]whereσis simple andm≥4, then theαis are unique.
The degenerate cases occur when a permutation can be written as an inflation of either 12or 21, and we may choose a unique decomposition in these cases in a variety of ways. The principal decomposition that we will use for the substitution decomposition, however, is as described in Proposition1.6.
The direct sum of two permutationsα and β is the inflation 12[α, β], and is usually denotedα⊕β. Similarly, theskew sumis the inflation21[α, β], and is denotedα β. The
direct sum operation acts as a dichotomy on the set of all permutations – dividing them into those that are sum decomposable (i.e. they can be represented as a direct sum), and those that are sum indecomposable. Similarly, the skew sum operation leads to theskew decomposablepermutations, while those that cannot be represented as a skew sum areskew indecomposable.
With these definitions, ifπcan be written as a direct sum (i.e. an inflation of the simple
permutation12), then we may writeπ = ιm[α1, . . . , αm] uniquely wherem is maximal,
and each αi is sum indecomposable. Similarly, if π is an inflation of21, we may write π =δm[α1, . . . , αm]where eachαi is skew indecomposable.
Alternatively, we may prefer to expressπ as the inflation of12or21, in which case we will specify which deflation we want; the one that follows will be the decomposition we
1.5 INFLATIONS AND THESUBSTITUTIONDECOMPOSITION 19 452398167 4523 45 4 5 23 2 3 98 9 8 1 67 6 7
Figure 1.6:The substitution decomposition tree ofπ= 452398167. mostly use.
Proposition 1.8(Albert and Atkinson [2]). Ifπis an inflation of12, then there is a unique sum
indecomposable α1 such that π = 12[α1, α2] for someα2, which is itself unique. The same holds
with12replaced by21and “sum” replaced by “skew”.
The substitution decomposition tree for a permutation then follows immediately. For example, consider the permutationπ= 452398167. This is decomposed as
452398167 = 2413[3412,21,1,12]
= 2413[21[12,12],21[1,1],1,12[1,1]]
= 2413[21[12[1,1],12[1,1]],21[1,1],1,12[1,1]] and its substitution decomposition tree is given in Figure 1.6.
CHAPTER
2
DECOMPOSITION
2.1 Background
S
INCE simple permutations may be used to construct all other permutations via thesubstitution decomposition, it would be useful to know how simple permutations are themselves constructed. In particular, our aim is to find smaller “fundamental” simple permutations of some specified size within a given simple permutation. Some approaches to this question can be found in Schmerl and Trotter [107], in which the following is proved for all irreflexive binary relational structures.1 Here, however, we will state only the per-
mutation case, for which there is another proof by Murphy [97].
Theorem 2.1(Schmerl and Trotter [107]). Every simple permutation of lengthn≥2contains a
simple permutation of lengthn−1orn−2.
We will prove that long simple permutations must contain two long almost disjoint simple subsequences. Formally:
Theorem 2.2. There is a functionf(k)such that every simple permutation of length at leastf(k)
contains two simple subsequences, each of length at leastk, sharing at most two entries.
(The proof of Theorem2.2follows after establishing Theorem2.14, found on Page34.) The second “two” in the statement of Theorem2.2is best possible, as is demonstrated by
1A version of this theorem fork-structures– structures defined on a singlek-ary relation in which every
relation(a1, . . . , ak)hasai=6 aj for somei6=j– can be found in Ehrenfeucht and McConnell [48].
Figure 2.1:The plots of a wedge simple permutation. Note that every simple subsequence of length at least4must contain its first two entries.
the family of simple permutations of the form
m(2m)(m−1)(m+ 1)(m−2)(m+ 2)· · ·1(2m−1);
the permutation in Figure2.1is of this form. On the other hand, no attempt has been made to optimise the functionf; our proof gives anf of order aboutkk.
This result alone, however, gives no real indication as to the underlying structure within the simple permutation; rather it is the method by which we arrive at Theorem2.2. We give a Ramsey-type description of simple permutations in terms of some unavoidable substructures, similar to the Erd˝os-Szekeres Theorem as applied to arbitrary permutations: Theorem 2.3 (Erd˝os and Szekeres [53]). Every permutation of length ncontains a monotone increasing or monotone decreasing subsequence of length at least √n.
In particular, we will demonstrate how a sufficiently long simple permutation contains, in the first instance, a “parallel alternation” of lengthk, a “wedge alternation” of lengthk
or a “pin sequence” of lengthk. By studying the decomposition of pin sequences, we can
go further to provide a more straightforward result, namely every sufficiently long simple permutation contains either an “alternation” or an “oscillation”.
A major motivation of this study is the enumeration of particular permutation classes. Although we will delay an in-depth discussion of this until Part II, it is worth noting that establishing a method of classifying the simple permutations brings us much closer to establishing what simple permutations lie in a given class.
2.2 PINSEQUENCES 23 p1 p2 p3 p1 p2 p3 p4 p1 p2 p3 p4 p5 p1 p2 p3 p4 p5 p6 p1 p2 p3 p4 p5 p6 p7 p1 p2 p3 p4 p5 p6 p7 p8
Figure 2.2:A pin sequence.