Homogeneous Permutations

There are just five Fraı̈ssé classes of permutations (apart from the trivial class of permutations of a singleton set); these are the identity permutations, reversing permutations, composites (in either order) of these two classes, and all permutations. The paper also discusses infinite generalisations of permutations, and the connection with Fraı̈ssé’s theory of countable homogeneous structures, and states a few open problems. Links with enumeration results, and the analogous result for circular permutations, are also described. 1 What is an infinite permutation? There are several ways of viewing a permutation of the finite set {1, . . . ,n}, giving rise to completely different infinite generalisations. To an algebraist, a permutation is a bijective mapping from X to itself. This definition immediately extends to an arbitrary set. The set of all permutations of any set X is a group under composition, the symmetric group Sym(X). A combinatorialist regards a permutation of {1, . . . ,n} in passive form, as the elements of {1, . . . ,n} arranged in a sequence (a1,a2, . . . ,an). If we try to extend this definition to the infinite, we are immediately faced with a problem: what kind of sequence should we use? For example, should it be well-ordered? A more satisfactory approach is to regard a permutation of {1, . . . ,n} as a pair of total orders, where the first is the natural order and the second is the order a1 < a2 < · · · < an of the terms in the sequence. Thus a permutation is a relational structure over the language with two binary relational symbols (interpreted as total orders). In this aspect, the infinite generalisation is clear, but the result is different from the other two. On an infinite set X , a pair of total orders do not correspond to a single permutation, but to a double coset G1πG2 in Sym(X), where G1 and G2 are the automorphism groups of the two the electronic journal of combinatorics 9 (2) (2002), #R2 1 total orders. (In the finite case, of course, a total order is rigid, so this double coset contains just the single permutation π.) This representation also makes the notion of subpermutation clear; it is simply the induced substructure on a subset Y of X (the restriction of the two total orders to Y ). I will adopt this view of permutations here. Accordingly, a finite permutation will be regarded as a pair of total orders, each represented by a sequence. For example, the permutation usually written in passive form as (2,4,1,3) might be represented as (abcd,bdac). I will call 2413 the pattern of this structure. Thus, a finite permutation is the pattern of an isomorphism class of finite structures (each consisting of a set with two total orders). The two total orders are denoted <1 and <2. 2 Ages and amalgamation A relational structure X is homogeneous if any isomorphism between finite substructures of X can be extended to an automorphism of X . The age of a relational structure X is the class of all finite structures embeddable in X . The best-known homogeneous structure is the ordered set Q. Fraı̈ssé [8], taking this as a prototype, gave a necessary and sufficient condition for a class of finite structures to be the age of a countable homogeneous relational structure. The four conditions are listed below; a class C of finite structures satisfying them is called a Fraı̈ssé class. (a) C is closed under isomorphism. (b) C is closed under taking induced substructures. (c) C has only countably many members (up to isomorphism). (d) C has the amalgamation property: if A,B1,B2 ∈ C and fi : A → Bi are embeddings for i = 1,2, then there exist C ∈ C and embeddings gi : Bi →C for i = 1,2 such that f1g1 = f2g2 (where f1g1 means the result of applying f1 and then g1). The amalgamation property informally says that two structures with a common substructure can be glued together. Fraı̈ssé further showed using a back-and-forth argument that, if C is a Fraı̈ssé class, then the countable homogeneous structure X whose age is C is unique up to isomorphism. We call X the Fraı̈ssé limit of C. Some authors (for example, Hodges [9]) include also the joint embedding property here. This is the following apparent weakening of the amalgamation property: given B1,B2 ∈ C, there exists C ∈ C such that both B1 and B2 can be embedded in C. These authors usually require a substructure to be non-empty; I will allow the empty structure (but assume that it is unique up to isomorphism). With this convention, the joint embedding property is a special case of the amalgamation property. It is easy to see that conditions (a)–(c) above and the joint embedding property are necessary and sufficient for C to be the age of some countable structure; but such a structure is by no means unique in general. See Hodges [9], Chapter 6, for further discussion of this material. the electronic journal of combinatorics 9 (2) (2002), #R2 2 Now we interpret (a)–(d) for the structures associated with permutations (sets with a pair of total orders). Since a pattern specifies an isomorphism class, (a) means that such a class is defined by a set C of patterns. Condition (b), called the hereditary property, of course means that C is defined by a set of excluded subpermutations. Condition (c) is vacuous. So the amalgamation property is the crucial condition. We will not always distinguish carefully between a class C of relational structures and the corresponding class C of permutations! The aim of this paper is to determine the Fraı̈ssé classes of permutations (and so, implicitly, the countable homogeneous structures consisting of a set with a pair of total orders). The classes will be described in the next section, and the theorem proved in the section following. Note that Murphy [12] has considered the question of hereditary classes of permutations with the joint embedding property (that is, ages of infinite permutations). Countable homogeneous graphs, digraphs and posets have been determined [10, 4, 13]. The result of this paper is analogous (though rather easier); but as far as I can see it does not follow from existing classifications. Much effort has been devoted to enumerating the permutations in various classes. In particular, the Stanley–Wilf conjecture [1] asserts that a hereditary class not containing all permutations has at most cn permutations on n points, for some constant c. On the other hand, Macpherson [11] showed that any primitive Fraı̈ssé class of relational structures of arbitrary signature (one whose members do not carry a natural equivalence relation derived from the structure) has at least cn/p(n) members of given cardinality, provided that it has more than one member of some cardinality. (Here c is an absolute constant greater than 1, and p a polynomial.) Examples where the growth is no faster than exponential are comparatively rare. It would appear that permutations would be a good place to look for such examples: this was part of the motivation for the present paper. From this point of view, the main theorem of this paper is a disappointment: of the five Fraı̈ssé classes of permutations defined below, J and J ∗ are trivial, J /J ∗ and J ∗/J are imprimitive, and U consists of all permutations.