Generating Functions to Know for Algebraic Combinatorics

Generating functions are powerful tools in combinatorics, helping to count and analyze sequences. Ordinary generating functions (OGFs) focus on selections without order, while exponential generating functions (EGFs) handle labeled structures, making them essential for various counting problems.

1. Ordinary generating functions (OGFs)

   • Defined as ( G(x) = a_0 + a_1 x + a_2 x^2 + \ldots ) where ( a_n ) represents the sequence of coefficients.
   • Useful for counting problems where the order of selection does not matter.
   • The coefficients of the series correspond to the number of ways to choose items.

2. Exponential generating functions (EGFs)

   • Defined as ( G(x) = a_0 + \frac{a_1 x}{1!} + \frac{a_2 x^2}{2!} + \ldots ) where ( a_n ) are the coefficients.
   • Particularly useful for counting labeled structures, such as permutations.
   • The factorial in the denominator accounts for the ordering of distinct objects.

3. Formal power series

   • A series of the form ( f(x) = a_0 + a_1 x + a_2 x^2 + \ldots ) without concern for convergence.
   • Provides a framework for manipulating sequences algebraically.
   • Can be used to represent generating functions and analyze their properties.

4. Convolution of sequences

   • The convolution of two sequences ( a_n ) and ( b_n ) is defined as ( c_n = \sum_{k=0}^{n} a_k b_{n-k} ).
   • Corresponds to the multiplication of their generating functions.
   • Useful in combinatorial problems involving combinations of independent choices.

5. Generating function manipulations (addition, multiplication, differentiation)

   • Addition of generating functions corresponds to the addition of sequences.
   • Multiplication of generating functions corresponds to the convolution of sequences.
   • Differentiation can be used to extract coefficients or to find relationships between sequences.

6. Closed forms of common generating functions

   • Certain sequences have well-known generating functions, such as ( G(x) = \frac{1}{1-x} ) for the sequence of natural numbers.
   • Closed forms simplify the analysis and computation of generating functions.
   • Important for recognizing patterns and deriving new results.

7. Partial fraction decomposition

   • A technique used to break down complex generating functions into simpler fractions.
   • Facilitates the extraction of coefficients and simplifies calculations.
   • Particularly useful for rational generating functions.

8. Coefficient extraction techniques

   • Methods such as the binomial theorem or Cauchy’s integral formula can be used to find specific coefficients.
   • Essential for solving combinatorial problems where specific terms are needed.
   • Techniques can include series expansion and manipulation of generating functions.

9. Recurrence relations and generating functions

   • Generating functions can be used to solve recurrence relations by transforming them into algebraic equations.
   • Provides a systematic way to find closed forms for sequences defined recursively.
   • Useful in deriving relationships between different combinatorial structures.

10. Solving linear recurrences using generating functions

    • Linear recurrences can be expressed in terms of their generating functions, allowing for algebraic manipulation.
    • Solutions can often be found by identifying patterns in the coefficients.
    • Provides a powerful tool for analyzing sequences defined by linear relationships.

11. Combinatorial applications of generating functions

    • Generating functions are used to solve counting problems, such as counting partitions or combinations.
    • They provide a unifying framework for various combinatorial identities and theorems.
    • Useful in enumerating structures in graph theory, number theory, and other areas.

12. Bivariate generating functions

    • Generating functions that depend on two variables, often used to encode two-dimensional counting problems.
    • Useful for studying relationships between two sequences or for counting pairs of objects.
    • Can be manipulated similarly to univariate generating functions.

13. Probability generating functions

    • A specific type of generating function used in probability theory to encode the probabilities of a random variable.
    • Defined as ( G(s) = E[s^X] ) where ( X ) is a random variable.
    • Useful for finding moments and analyzing distributions.

14. Moment generating functions

    • A type of generating function that encodes the moments of a probability distribution.
    • Defined as ( M(t) = E[e^{tX}] ) where ( X ) is a random variable.
    • Useful for characterizing distributions and finding relationships between random variables.

15. Generating functions for partitions

    • Generating functions can be used to study the number of ways to partition integers.
    • The partition function ( p(n) ) can be represented using generating functions.
    • Important in number theory and combinatorial analysis, providing insights into the structure of partitions.