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Unformatted text preview: 3. C LASSES In the previous two lectures, we have seen that the notion of permutation containment and avoidance naturally describes the permutations that can be sorted by simple sorting machines. Now we study permutation contain- ment in the abstract. 3.1. D EFINITIONS Let us write σ π if σ is contained in π . This relation is a partial order 1 on the set of all permutations. A permutation class (or simply class ) is a set of permutations which is closed downward under permutation containment. Thus if C is a permutation class and σ π for some π C , then σ also lies in C . (We use upper-case calligraphic letters for permutation classes throughout; sets of permutations which do not form classes are denoted by normal upper-case letters.) There are roughly three ways to describe a permutation class: • a class may be specified as all the permutations sortable by a particu- lar sorting machine, • a class may be specified by the minimal permutations it does not con- tain, or • a class may be specified by a certain structural property. We saw the first method in Lecture 1, where we considered the classes of permutations sortable with a stack or with two parallel queues. In an analogous manner, sorting machines can be used to generate permutations, as considered in Exercises 1 and 2 of Lecture 1. We have also seen the second method; for example the stack-sortable permutations can also be described as the 231-avoiding permutations (The- orem 1.3), and the permutations sortable with two parallel queues are the 321-avoiding permutations (Theorem 1.7). The manner we have specified these classes (that they contain all the permutations that avoid a given pat- tern) can be extended to all permutation classes: for every permutation class C there is a unique set B consisting of the minimal permutations which are not in C . We refer to B as the basis 2 of C . Conversely, any set B of permu- 1 Formally, this means that permutation containment is reflexive ( π π ), transitive (if τ σ and σ π then τ π ), and antisymmetric (if σ π and π σ then π σ ). 2 This terminology is standard, but unfortunate because in most settings the “basis” of an object generates the object in some sense, whereas for permutation classes the relationship is reversed. 13 14 LECTURE 3 C LASSES tations determines a class of permutations, for which we use the notation Av B : Av B π : β π for all β B . Note that no permutation in the basis of a class can be contained in another member of the basis, because the basis elements must be minimal. This means that the basis is an antichain . Infinite antichains (and hence infinite bases) of permutations do exist, as demonstrated by Exercise 14....
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- Spring '11
- Generating function, π, gπ