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By Flajolet P., Sedgewick R.

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Since we restrict attention to admissible constructions, we can immediately derive OGFs for these classes. Put differently, the task of enumerating a combinatorial class is reduced to programming a specification for it in the language of admissible constructions. 2. First, in the framework just introduced, the class of all binary words is described by W = S EQ(A) where A = {a, b} ∼ = Z + Z, the ground alphabet, comprises two elements (letters) of size 1. The size of a binary word then coincides with its length (the number of letters it contains).

40), the OGF of Fibonacci numbers. 26 I. UNLABELLED STRUCTURES AND ORDINARY GENERATING FUNCTIONS I. 2. The admissibility theorem for ordinary generating functions. This section is a formal treatment of admissibility proofs for the constructions we have considered. The final implication is that any specification of a constructible class translates directly into generating function equations. The cycle construction involves the Euler totient function ϕ(k) defined as the number of integers in [1, k] that are relatively prime to k (A PPENDIX A: Arithmetical functions, p.

PN can be computed in O(N 2 ) integer-arithmetic operations. (The technique is generally applicable to powersets and multisets; see Note √ 40 for another application. ) By varying (27) and (28), we can use the symbolic method to derive a number of counting results in a straightforward manner. 1. Let T ⊆ I be a subset of the positive integers. The OGF of the classes C T := S EQ(S EQ T (Z)) and P T := MS ET(S EQ T (Z)) of compositions and partitions having summands restricted to T is given by 1 1 1 = .

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