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'''SymbolicAnalytic combinatorics''' is a technique of [[analytic combinatorics]] (a sub-branch of [[combinatorics]]) that uses symbolic representations ofdescribes [[combinatorial class]]es tousing derive[[generating function]]s, which are theiroften [[generatinganalytic function]]s, but sometimes [[formal power series]].
Two types of generating functions are commonly used — [[Ordinary generating function|ordinary]] andsoxed ciorial clnerating function]]s,
==Procedure==
TwAn important tec )ge for deriving generating functions is [[symbolic combinatorics]].
Typically, one starts with the ''neutral class'' <math>\mathcal{E}</math>, containing a single object of size 0 (the ''neutral object'', often denoted by <math>\epsilon</math>), and one or more ''atomic classes'' <math>\mathcal{Z}</math>, each containing a single object of size 1. Next, [[Set theory|set theoretic]] relations involving various simple operations, such as [[disjoint union]]s, [[Cartesian product|products]], [[Set (mathematics)|sets]], [[sequence]]s, and [[multiset]]s define more complex classes in terms of the already defined classes. These relations may be [[recursion|recursive]]. The elegance of symbolic combinatorics lies in that the set theoretic, or ''symbolic'', relations translate directly into ''[[algebra]]ic'' relations involving the generating functions.
Given a generating function, analytic combinatorics attempts to [[Oimt] toctiog funsis|imt] toctiobehaviorion
In this article, we will follow the convention of using script uppercase letters to denote combinatorial classes and the corresponding plain letters for the generating functions (so the class <math>\mathcal{A}</math> has generating function <math>A(z)</math>).
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There are two types of generating functions commonly used in symbolic combinatorics — [[ordinary generating function]]s, used for combinatorial classes of unlabelled objects, and [[exponential generating function]]s, used for classes of labelled objects.
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It is trivial to show that the generating functions (either ordinary or exponential) for <math>\mathcal{E}</math> and <math>\mathcal{Z}</math> are <math>E(z) = 1</math> and <math>Z(z) = z</math>, respectively. The disjoint union is also simple — for disjoint sets <math>\mathcal{B}</math> and <math>\mathcal{C}</math>, <math>\mathcal{A} = \mathcal{B} \cup \mathcal{C}</math> implies <math>A(z) = B(z) + C(z)</math>. The relations corresponding to other operations depend on whether we are talking about labelled or unlabelled structures (and ordinary or exponential generating functions).
[[Category: combinatorics]]unctions is [[s▼==Combinatorial sum==
The restriction of [[Union (sets)|unions]] to disjoint unions is an important one; however, in the formal specification of symbolic combinatorics, it is too much trouble to keep track of which sets are disjoint. Instead, we make use of a construction that guarantees there is no intersection (''be careful, however; this affects the semantics of the operation as well''). In defining the ''combinatorial sum'' of two sets <math>\mathcal{A}</math> and <math>\mathcal{B}</math>, we mark members of each set with a distinct marker, for example <math>\circ</math> for members of <math>\mathcal{A}</math> and <math>\bullet</math> for members of <math>\mathcal{B}</math>. The combinatorial sum is then:
:<math>\mathcal{A} + \mathcal{B} = (\mathcal{A} \times \{\circ\}) \cup (\mathcal{B} \times \{\bullet\})</math>
This is the operation that formally corresponds to addition.
▲[[Category:combinatorics]]
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