Combinatorics

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By Flajolet P., Sedgewick R.

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Distinct copies of the neutral or atomic class may also be subscripted by indices in various ways. Thus, for instance we use the classes Za = {a}, Zb = {b} (with a, b of size 1) to build up binary words over the alphabet {a, b}, or Z• = {•}, Z◦ = {◦} (with •, ◦ taken to be of size 1) to build trees with nodes of two coulurs. Similarly, we introduce E , E1 , E2 to denote a class comprising the neutral objects , ǫ1 , ǫ2 respectively. Clearly, the generating functions of a neutral class E and an atomic class Z are E(z) = 1, Z(z) = z, corresponding to the unit 1, and the variable z, of generating functions.

N + 1)! n! with OGF C(z) = 1− √ 1 − 4z 2z are known as the Catalan numbers (EIS A000108) in the honour of Eug`ene Catalan (1814-1894), a French and Belgian mathematician who developed many of their properties. These numbers are so common in combinatorics that we have decided to use a roman font for denoting them (like “log”, “sin”, and so on). In summary, general trees are enumerated by Catalan numbers: Gn = Cn−1 ≡ 1 2n − 2 . n n−1 For this reason the term Catalan tree is often employed as synonymous to “general (rooted unlabelled plane) tree”.

Consequently, P1 , . . , PN can be computed in O(N 2 ) integer-arithmetic operations. (The technique is generally applicable to powersets and multisets; see Note √ 40 for another application. ) By varying (27) and (28), we can use the symbolic method to derive a number of counting results in a straightforward manner. 1. Let T ⊆ I be a subset of the positive integers. The OGF of the classes C T := S EQ(S EQ T (Z)) and P T := MS ET(S EQ T (Z)) of compositions and partitions having summands restricted to T is given by 1 1 1 = .

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