By Flajolet P., Sedgewick R.

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In other words, we have A = (β1 , . . , βℓ ) ℓ ≥ 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 I. 2. ADMISSIBLE CONSTRUCTIONS AND SPECIFICATIONS 25 follows that the size of an object α ∈ A is to be taken as the sum of the sizes of its components: α = (β1 , .

The classification of growth rates of counting sequences belongs properly to the asymptotic theory of combinatorial structures which neatly relates to the symbolic method via complex analysis. A thorough treatment of this part of the theory is presented in Chapters IV–VIII. Given the methods expounded there, it becomes possible to estimate asymptotically the coefficients of virtually any generating function, however complicated, that is provided by the symbolic method; that is, implicit enumerations in the sense above are well covered by complex asymptotic methods.

In particular, k times S EQk (B) := B × · · · × B ≡ B k , S EQ≥k (B) = j≥k Bj ∼ = B k × S EQ(B), MS ETk (B) := S EQk (B)/R. Similarly, S EQodd , S EQeven will denote sequences with an odd or even number of components, and so on. 30 I. COMBINATORIAL STRUCTURES AND ORDINARY GENERATING FUNCTIONS Translations for such restricted constructions are available, as shown generally in Subsection I. 1, p. 82. Suffice it to note for the moment that the construction A = S EQk (B) is really an abbreviation for a k-fold product, hence it admits the translation into OGFs (24) A = S EQk (B) ⇒ A(z) = B(z)k .