By Flajolet P., Sedgewick R.

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1 2 3 (See A PPENDIX A: Formal power series, p. ) Multiset construction. First for finite B (with B0 = ∅), the multiset class A = MS ET(B) is definable by MS ET(B) ∼ = S EQ(β). β∈B In words, any multiset can be sorted, in which case it can be viewed as formed of a sequence of repeated elements β1 , followed by a sequence of repeated elements β2 , where β1 , β2 , . . is a canonical listing of the elements of B. The relation translates into generating functions by the product and sequence rules, A(z) = β∈B = exp (1 − z |β|)−1 = ∞ n=1 ∞ (1 − z n )−Bn n=1 Bn log(1 − z n )−1 B(z) B(z 2 ) B(z 3 ) + + + ··· , 1 2 3 where the exponential form results from the exp-log transformation.

In fact, it is even possible to use computer algebra systems in order to compute it automatically! See the article of Flajolet, Salvy, and Zimmermann [159] for the description of such a system. 2 (Symbolic method, unlabelled case). The generating function of a constructible class is a component of a system of generating function equations whose terms are built from 1, z, + , × , Q , Exp , Exp , Log, where Q[f ] = Exp = 1 , 1−f exp ∞ k=1 Log[f ] = ∞ k=1 f (z k ) k , Exp[f ] = exp 1 ϕ(k) log , k 1 − f (z k ) ∞ k=1 (−1)k−1 f (z k ) k .

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.