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# 数学代写|组合学代写Combinatorics代考|MATH4410 Exploiting generating functions and counting sequence

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## 数学代写|组合学代写Combinatorics代考|Exploiting generating functions and counting sequence

Exploiting generating functions and counting sequences. In this book we are going to see altogether more than a hundred applications of the symbolic method. Before engaging in technical developments, it is worth inserting a few comments on the way generating functions and counting sequences can be put to good use in order to solve combinatorial problems.

Explicit enumeration formuld. In a number of situations, generating functions are explicit and can be expanded in such a way that explicit formulae result for their coefficients. A prime example is the counting of general trees and of triangulations above, where the quadratic equation satisfied by an OGF is amenable to an explicit solution-the resulting OGF could then be expanded by means of Newton’s binomial theorem. Similarly, we derive later in this Chapter an explicit form for the number of integer compositions by means of the symbolic method and OGFs the answer turns out to be simply $2^{n-1}$ ) and derive many explicit specializations. In this book, we assume as known the elementary techniques from basic calculus by which the Taylor expansion of an explicitly given function can be obtained. Good references on such elementary aspects are Wilf’s Generatingfunctionology [406], Graham, Knuth, and Patashnik’s Concrete Mathematics [196], and our book [353].

## 数学代写|组合学代写Combinatorics代考|Implicit enumeration formulæ

Implicit enumeration formula. In a number of cases, the generating functions obtained by the symbolic method are still in a sense explicit, but their form is such that their coefficients are not clearly reducible to a closed form. It is then still possible to obtain initial values of the corresponding counting sequence by means of a symbolic manipulation system. Also, from generating functions, it is possible to derive systematically recurrences $^2$ that lead to a procedure for computing an arbitrary number of terms of the counting sequence in a reasonably efficient manner. A typical example of this situation is the OGF of integer partitions,
$$P(z)=\prod_{m=1}^{\infty} \frac{1}{1-z^m},$$
for which recurrences obtained from the $\mathrm{OGF}$ and associated to fast algorithms are given in Note 12 (p. 39) and Note 17 (p. 46).

## 数学代写|组合学代写Combinatorics代考|Implicit enumeration formulæ

$$P(z)=\prod_{m=1}^{\infty} \frac{1}{1-z^m}$$

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