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We easily obtain cn ∼ n! as n→∞, so that almost all permutations are indecomposable. 18) Some further useful expansions for functional inverses and computations of formal power series have been obtained by Bender and Richmond . 8. Elementary estimates for convergent generating functions The word “elementary” in the title of this section is a technical term that means the proofs do not use complex variables. It does not necessarily imply that the proofs are simple. 1, are easy, others are more complicated.
1 to multivariate functions can also be derived, but are again harder than the upper bound . 2. 1.
The helpful feature of the inequalities is that they require only knowledge of the pairwise dependencies among the Y j , which is easier to study than the full joint distribution of the Y j . For other bounds on distributions that can be obtained from partial information about moments, see . The reason moment bounds are mentioned at all in this chapter is that asymptotic methods are often used to derive them. Generating functions are a common and convenient method for doing this. 8. Waiting times for subwords.
Asymptotic enumeration methods by Odlyzko.