Combinatorics

# Download Analytic combinatorics MAc by Flajolet P., Sedgewick R. PDF

By Flajolet P., Sedgewick R.

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Extra resources for Analytic combinatorics MAc

Example text

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.