Part II Permutation Classes
5.5 The Containment Partial Order in Other Structures
We may, of course, define the containment order on any relational structure and treat it as a partial order. Expanding upon the notion of extensions in Chapter 3, ifAandBare
relational structures over a common languageLthen anembeddingofAintoBis an injec-
tionϕ: dom(A) → dom(B)so thatB|ϕ(dom(A)) is isomorphic toA. If such an embedding exists, then we say A ≤ B, a quasi order from which we may induce a partial order by
considering the equivalence classesA ∼=B, arising if and only ifA ≤ BandB ≤ A.
In theory, one may then study any closed class of relational structures for a given lan- guage in the same way as one might study permutation classes. Formally, a setCof rela- tional structures over a common relational languageL is anL-class ifA ∈ C andB ≤ A impliesB ∈C. We might then if we wished define anL-class in terms of structure avoid- ance and try to compute its generating function. We could consider intersections, unions and, by recalling the definition of inflation in this general setting, wreath products and wreath closures.
Antichains, partial well order and atomicity are notions taken from the theory of posets. Antichains are merely sets of pairwise incomparable elements; see Gustedt [66] for notions of minimality in antichains and some considerations on the existence of infinite antichains. Since everyL-class has a minimal element on one point, noL-class can contain an infinite
properly decreasing sequence. Thus an L-classC ispartially well ordered if it contains no infinite antichains, and Higman’s Theorem can be used in the general setting. Atomicity in the permutation class case is merely a special case of the “γ classes” of Fra¨ıss´e [56]; many
of the results that are true for permutation classes are also true in the general case. For example, an atomicL-classCsatisfies the joint embedding property, and is also expressible in a way analogous to theSub(π :A → B)notation. See Fra¨ıss´e [56], Hodges [68, Section 7.1], and, for a survey of more recent results, Pouzet [101].
Finitely many Simples. By means of the substitution decomposition, L-classes which
contain only finitely many simpleL-structures will have a recursive construction much as
in the permutation class case. However in the general setting this does not correspond to an algebraic generating function, since structures in the partial order are defined only up to equivalence. In fact, it seems that having an algebraic generating function is special to the permutation case (for example, it is not true in the graph case).
All suchL-classes are, however, partially well ordered. As in the permutation case,
antichains are instrinsically linked to simple permutations, and Proposition5.27is proved in the general case by Gustedt [66].
To answer the question of whether these classes are finitely based, we may obtain a partial answer by considering the most general setting of the Schmerl-Trotter Theorem 2.1
given in [107], namely that of binary, irreflexive relational structures, a set which includes graphs, tournaments and posets. Ehrenfeucht and McConnell [48] show that, fork ≥ 3, a simple structure defined on a single k-ary relation must contain a simple substructure
withk,k−1ork−2fewer points, and this was improved to justk−1ork−2fewer points by Bonizzoni and McConnell [23]. Further generalisations remain unknown.
The Graph Case. The “graph containment order” is in fact the order defined by induced subgraphs, and has been extensively studied. As with many other relational structures, classes of graphs closed under taking induced subgraphs are more often referred to as
hereditary properties. A stronger condition is obtained by considering sets of graphs closed under taking subgraphs (rather than induced subgraphs), and these are referred to as
monotoneproperties.
5.5 THECONTAINMENT PARTIALORDER IN OTHERSTRUCTURES 109
regular graphs. Examples of hereditary properties include the set of triangle-free graphs, all graphs of chromatic number at mostkand the set of split graphs (graphs which may be
partitioned into an independent set and a clique).
As with permutation classes, much of the study of hereditary graph properties is in their asymptotic enumeration. For a propertyP, letPndenote the set of graphs inP with nvertices, whence the function|Pn| defines thespeed of the property. While little can be
said about the speed of an arbitrary property, Scheinerman and Zito [106] prove that the speed of hereditary graph properties must, for sufficiently large n, be constant, polyno-
mial, exponential, factorial or superfactorial. Subsequent study – in particular Balogh, Bollob´as and Weinreich [15,16] – has shown that there are many “jumps” within this al- ready broken spectrum of speeds.
CHAPTER
6
ALGEBRAIC
GENERATING
FUNCTIONS
6.1 Introduction
W
HEN A CLASS is enumerated by an algebraic generating function, we intuitivelyexpect to find some recursive description of the permutations in the class. Such de- scriptions may arise in a variety of ways, but one of the most important is the substitution decomposition.
In a class which has only finitely many simple permutations, therefore, any long per- mutation must map nontrivial intervals onto intervals, and hence all the permutations of the class are constructed recursively via the substitution decomposition. With only finitely many simple permutations on which to “build”, we expect the class to have an algebraic generating function:
Theorem 6.1 (Albert and Atkinson [2]). A permutation class with only finitely many simple permutations has a readily computable algebraic generating function.
Our aim in this chapter is to establish a generalisation of Theorem6.1. We do this by observing that the recursive construction given by the substitution decomposition is not a feature merely of pattern avoidance in the containment order, but can be extended to enumerate a wide variety of other sets of permutations. In essence it can be extended to enumerate any set of permutations which can be built in the same way from a finite set of simple permutations, though we will still require that the set lies within a permutation class with only finitely many simple permutations.
Theorem 6.2. Let C be a permutation class containing only finitely many simple permutations, P a finite query-complete set of properties, and Q ⊆ P. The generating function for the set of
permutations inCsatisfying every property inQis algebraic overQ(x).
The next section establishes the terminology required by Theorem6.2, which we will then prove in Section 6.3. Section 6.4 shows how to describe some common families of permutations as query-complete sets of properties and hence demonstrates the scope of Theorem6.2, with specific worked examples given in Section6.5. In Sections6.6and6.7we adapt these techniques to enumerate two further families, namely involutions and cyclic closures, respectively. Some closing remarks are given in Section 6.8.