By Flajolet P., Sedgewick R.

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One good reason behind the definition adopted here is that the combinatorial sum of two classes is always well defined. Furthermore, disjoint union is equivalent to a standard union whenever it is applied to disjoint sets. Because of disjointness, one has the implication A=B+C ⇒ An = Bn + Cn and A(z) = B(z) + C(z), so that disjoint union is admissible. , the nature of this intersection) is needed in order to be able to enumerate the elements of their union. Cartesian product. 6. The size of a pair is obtained additively from the size of components in accordance with (9).

We next examine compositions and partitions with a fixed number of summands. 6. Compositions with a fixed number of parts. Let C (k) denote the class of compositions made of k summands, k a fixed integer ≥ 1. 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 ] zk n−1 , = k−1 (1 − z)k a result which constitutes a combinatorial refinement of Cn = 2n−1 .

6. Interval coverings. Let Z := {•} be as before. Then A = Z + (Z × Z) is a set of two elements, • and (•, •), which we choose to draw as {•, •–•}. Then C = S EQ(A) contains objects like •, • •, •–•, • •–•, •–• •, •–• •–•, • • • • . With the notion of size adopted, the objects of size n in C = S EQ(Z + (Z × Z)) are (isomorphic to) the coverings of the interval [0, n] by intervals (matches) of length either 1 or 2. The generating function C(z) = 1 + z + 2 z 2 + 3 z 3 + 5 z 4 + 8 z 5 + 13 z 6 + 21 z 7 + 34 z 8 + 55 z 9 + · · · , is, as we shall see shortly (p.