By Flajolet P., Sedgewick R.

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One has C (k) (z) = S EQk (I) ≡ I × I × · · · × I, where the number of terms in the cartesian product is k. From here, the corresponding generating function is found to be z k C (k) = I (z) with I (z) = . 1−z The number of compositions of n having k parts is thus (k) Cn = [z n ] n−1 zk = , k−1 (1 − z)k a result which constitutes a combinatorial refinement of Cn = 2n−1 . 6)). (k) In such a case, the asymptotic estimate Cn ∼ n k−1 /(k − 1)! results immediately from the polynomial form of the binomial coefficient n−1 k−1 .

A(r ) ) of classes is a collection of r equations, (1) (1) (r ) 1 (A , . . , A ) A(2) = (1) (r ) A = 2 (A , . . , A ) (26) · · · (r ) (1) (r ) A = r (A , . . , A ) where each i denotes a term built from the A using the constructions of disjoint union, cartesian product, sequence, set, multiset, and cycle, as well as the initial classes E (neutral) and Z (atomic). We also say that the system is a specification of A(1) . A specification for a combinatorial class is thus a sort of formal grammar defining that class.

The general argument also gives the generating function of partitions whose summands lie in the set {1, 2, . . ,r } (z) = r m=1 1 . 1 − zm I. 3. INTEGER COMPOSITIONS AND PARTITIONS 43 In other words, we are enumerating partitions according to the value of the largest summand. 9, p. ,r } (42) Pn ∼ cr nr −1 with cr = 1 . (r − 1)! A similar argument provides the asymptotic form of PnT when T is an arbitrary finite set: PnT ∼ 1 nr −1 τ (r − 1)! with τ := n, r := card(T ). 2, p. 257. . . . . .