By Robert Sedgewick, Philippe Flajolet

Analytic Combinatorics is a self-contained therapy of the maths underlying the research of discrete constructions, which has emerged during the last numerous many years as a vital software within the figuring out of homes of laptop courses and clinical versions with purposes in physics, biology and chemistry. Thorough therapy of a giant variety of classical purposes is an important point of the presentation. Written by means of the leaders within the box of analytic combinatorics, this article is bound to turn into the definitive reference at the subject. The textual content is complemented with routines, examples, appendices and notes to help realizing for that reason, it may be used because the foundation for a complicated undergraduate or a graduate direction at the topic, or for self-study.

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**Example text**

An axiomatic presentation is then as follows: a combinatorial class is a pair (A, | · |) where A is at most denumerable and the mapping | · | ∈ (A → Z≥0 ) is such that the inverse image of any integer is finite. 2. The counting sequence of a combinatorial class is the sequence of integers (An )n≥0 where An = card(An ) is the number of objects in class A that have size n. 1. Binary words. Consider first the set W of binary words, which are sequences of elements taken from the binary alphabet A = {0,1}, W := {ε, 0, 1, 00, 01, 10, 11, 000, 001, 010, .

In other terms, we start from basic atomic elements and build up words by forming freely all the objects determined by the sequence construction. Such a combinatorial description of a class that only involves a composition of basic constructions applied to initial classes E, Z is said to be an iterative (or non-recursive) specification. 1, p. 5, p. 27) respectively defined by N = C YC(Z + Z) and I = S EQ≥1 (Z). From this, one can construct ever more complicated objects. For instance, P = MS ET(I) ≡ MS ET(S EQ≥1 (Z)) means the class of multisets of positive integers, which is isomorphic to the class of integer partitions (see Section I.

0, β j ∈ B , which matches our intuition as to what sequences should be. ) It is then readily checked that the construction A = S EQ(B) defines a proper class satisfying the finiteness condition for sizes if and only if B contains no object of size 0. From the definition of size for sums and products, it 26 I. COMBINATORIAL STRUCTURES AND ORDINARY GENERATING FUNCTIONS follows that the size of an object α ∈ A is to be taken as the sum of the sizes of its components: α = (β1 , . . , β ) ⇒ |α| = |β1 | + · · · + |β |.