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Unformatted text preview: 6. A SYMPTOTICS So far we have only discussed the exact enumeration of permutation classes. Frequently, we are more interested in the asymptotics of a class, either be- cause we cannot obtain the exact enumeration, or because we only want a rough measure of size of the class. This is the topic of the current lecture, which takes us every so slightly into elementary analysis. 6.1. G ROWTH R ATES We saw in Lecture 2 that the classes Av 231 and Av 321 are both counted by the Catalan numbers, C n 1 n 1 2 n n . While this exact formula is nice, it doesn’t do a great job of telling us how large these numbers are. Obtaining a crude estimate is not very difficult. For an upper bound, note that the sum of all entries in the 2 n th row of Pascal’s Triangle is 2 2 n 4 n . For a lower bound, note that 2 n n is the largest entry in the 2 n th row of Pascal’s Triangle, so it must be at least the average, 4 n 2 n . This gives us 4 n n 1 2 n C n 4 n n 1 . Therefore we are justified in saying that C n grows like 4 n . In particular, almost all 1 permutations contain 231 . Now let C be any permutation class. If the limit exists, we say that lim n n C n lim n n the number of permutations of length n in C 1 This means that as n , the probability that a permutation of length n selected uniformly at random contains 231 tends to 1 , and should not be a surprise. Let β be any permutation of length k and choose π uniformly at random from among all permutations of length n . The probability that any specific k entries of π are in the same relative order as β is 1 k ! . Therefore if n kq r , we can group the first k entries of π together, the next k entries of π together, and so on, to obtain the bound Pr π avoids β 1 1 k ! q , which tends to as n . Therefore the probability that π contains β tends to 1 and n . 29 30 LECTURE 6 A SYMPTOTICS is the growth rate of C , and denote it by gr C . Our upcoming Theorem 6.1 provides one sufficient condition for the existence of growth rates, but it is not known if this limit exists for all permutation classes. Therefore we must in general make do with upper growth rates , gr C lim sup n n C n , and lower growth rates , gr C lim inf n n C n ....
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- Spring '11
- Trigraph, Supremum, CN