By Robert Sedgewick, Philippe Flajolet

Analytic Combinatorics is a self-contained therapy of the math underlying the research of discrete constructions, which has emerged over the last numerous many years as an important software within the knowing of houses of machine courses and medical types with purposes in physics, biology and chemistry. Thorough therapy of a big variety of classical purposes is a necessary element of the presentation. Written through the leaders within the box of analytic combinatorics, this article is sure to turn into the definitive reference at the subject. The textual content is complemented with workouts, examples, appendices and notes to help knowing accordingly, it may be used because the foundation for a complicated undergraduate or a graduate path at the topic, or for self-study.

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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. 3 below for a detailed discussion).

The notation is A = MS ET(B) when A is obtained by forming all finite multisets of elements from B. The precise way of defining MS ET(B) is as a quotient: MS ET(B) := S EQ(B)/R with R, the equivalence relation of sequences being defined by (α1 , . . , αr ) R (β1 , . . , βr ) iff there exists some arbitrary permutation σ of [1 . r ] such that for all j, β j = ασ ( j) . Powerset construction. The powerset class (or set class) A = PS ET(B) is defined as the class consisting of all finite subsets of class B, or equivalently, as the class PS ET(B) ⊂ MS ET(B) formed of multisets that involve no repetitions.

COMBINATORIAL STRUCTURES AND ORDINARY GENERATING FUNCTIONS where the exponential form results from the exp–log transformation. The case of an infinite class B follows by a limit argument analogous the one used for powersets. Cycle construction. The translation of the cycle relation A = C YC(B) turns out to be ∞ 1 ϕ(k) log , A(z) = k 1 − B(z k ) k=1 where ϕ(k) is the Euler totient function. The first terms, with L k (z) := log(1 − B(z k ))−1 are 1 1 2 2 4 2 A(z) = L 1 (z) + L 2 (z) + L 3 (z) + L 4 (z) + L 5 (z) + L 6 (z) + · · · .

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