Analytic combinatorics is a branch of combinatorics

Combinatorics

Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...

 that describes combinatorial class

Combinatorial class

In combinatorics, a combinatorial class is an equivalence class of sets that have the same counting sequence. Although the elements of these equivalent sets may have very different definitions and semantics, combinatorics is concerned only with the number of elements of a given size...

es using generating function

Generating function

In mathematics, a generating function is a formal power series in one indeterminate, whose coefficients encode information about a sequence of numbers an that is indexed by the natural numbers. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general...

s, with formal power series

Formal power series

In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates...

 that often correspond to analytic function

Analytic function

In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...

s.

Given a generating function, analytic combinatorics attempts to describe the asymptotic behavior

Asymptotic analysis

In mathematical analysis, asymptotic analysis is a method of describing limiting behavior. The methodology has applications across science. Examples are...

 of a counting sequence using algebra

Algebra

Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...

ic techniques. This often involves analysis of the singularities of the associated analytic function.

Two types of generating functions are commonly used — ordinary and exponential generating functions.

An important technique for deriving generating functions is symbolic combinatorics

Symbolic combinatorics

In mathematics, symbolic combinatorics is one of the many techniques of counting combinatorial objects. It uses the internal structure of the objects to derive formulas for their generating functions. This particular theory is due to Philippe Flajolet and Robert Sedgewick. This article describes...

.