2.1 Ordinary Generating Functions

Ordinary generating functions are powerful tools for solving combinatorial problems. They encode information about sequences into formal power series, allowing us to manipulate and analyze complex structures with simple algebraic operations.

These functions are crucial in the study of generating functions and recurrence relations. By representing combinatorial objects as coefficients in a series, we can uncover patterns, derive closed-form expressions, and explore connections between seemingly unrelated sequences.

Ordinary generating functions

Definition and properties

Top images from around the web for Definition and properties

• 

  Generating function - Wikipedia, the free encyclopedia View original

  Is this image relevant?

• 

  Generating function - Wikipedia, the free encyclopedia View original

  Is this image relevant?

1 of 1

Top images from around the web for Definition and properties

• 

  Generating function - Wikipedia, the free encyclopedia View original

  Is this image relevant?

• 

  Generating function - Wikipedia, the free encyclopedia View original

  Is this image relevant?

1 of 1

• An ordinary generating function is a formal power series where the coefficient of the nth term represents the number of ways to construct an object of size n
• Ordinary generating functions have the form $G(x) = a_0 + a_1x + a_2x^2 + ... = \sum_{n=0}^{\infty} a_n x^n$, where $a_n$ is the number of objects of size $n$
• The coefficient $a_n$ in an ordinary generating function can be extracted using the formula $a_n = [x^n]G(x)$, representing the coefficient of $x^n$ in $G(x)$
• Ordinary generating functions encode information about the number of objects of different sizes into a formal power series to solve combinatorial problems
• The multiplication of two ordinary generating functions corresponds to taking the Cartesian product of the associated combinatorial classes (generating functions for binary strings and integer partitions)

Applications and examples

• Ordinary generating functions can be used to solve problems related to counting objects with specific properties or satisfying certain constraints
• Examples of combinatorial objects that can be studied using ordinary generating functions include permutations, combinations, partitions, and lattice paths
• Ordinary generating functions can be used to analyze the asymptotic behavior of combinatorial sequences by studying the singularities and growth rates of the associated generating functions
• Ordinary generating functions can be used to derive closed-form expressions for combinatorial sequences by manipulating the generating function and extracting coefficients (Catalan numbers, Fibonacci numbers)
• Ordinary generating functions can be used to establish bijections between combinatorial objects by showing that their generating functions are equal (Dyck paths and valid parenthesizations)

Generating functions for sequences

Common combinatorial sequences

• The generating function for the sequence {1, 1, 1, ...} is $\frac{1}{1-x}$, representing the number of ways to select any number of identical objects
• The generating function for the sequence {1, 2, 3, ...} is $\frac{1}{(1-x)^2}$, representing the number of ways to select a multiset of objects from two distinct types
• The generating function for the sequence {1, 3, 6, 10, ...} (triangular numbers) is $\frac{1}{(1-x)^3}$, representing the number of ways to select a multiset of objects from three distinct types
• The generating function for the Fibonacci sequence {1, 1, 2, 3, 5, 8, ...} is $\frac{x}{1-x-x^2}$, obtained by solving a recurrence relation using generating functions
• The generating function for the Catalan numbers {1, 1, 2, 5, 14, 42, ...} is $\frac{1 - \sqrt{1-4x}}{2x}$, representing the number of valid parenthesizations of $n$ pairs of parentheses

Constructing generating functions

• Generating functions can be constructed by encoding the combinatorial structure of a sequence into a formal power series
• The coefficients of the generating function correspond to the number of objects of each size in the sequence
• Recurrence relations satisfied by a sequence can be translated into equations involving generating functions, which can then be solved to obtain the generating function (Fibonacci numbers, Catalan numbers)
• Combinatorial arguments can be used to derive the generating function for a sequence by considering the ways in which objects of different sizes can be constructed (binary trees, permutations with forbidden patterns)
• The generating function for a sequence can sometimes be guessed by recognizing patterns in the sequence and matching them to known generating functions (derangements, partition numbers)

Operations on generating functions

Arithmetic operations

• The sum of two generating functions $G(x)$ and $H(x)$ is $(G+H)(x) = G(x) + H(x)$, corresponding to the disjoint union of the associated combinatorial classes
• The product of two generating functions $G(x)$ and $H(x)$ is $(G*H)(x) = G(x) * H(x)$, corresponding to the Cartesian product of the associated combinatorial classes
• The composition of two generating functions $G(x)$ and $H(x)$ is $(G \circ H)(x) = G(H(x))$, corresponding to the substitution of combinatorial structures
• The quotient of two generating functions $G(x)$ and $H(x)$ is $(\frac{G}{H})(x) = G(x) / H(x)$, which can be used to solve certain types of combinatorial problems (partitions into distinct parts, surjections)
• The generating function for the sequence obtained by convolving two sequences with generating functions $G(x)$ and $H(x)$ is the product $G(x) * H(x)$ (Vandermonde convolution)

Calculus operations

• The derivative of a generating function $G(x)$ is $G'(x) = a_1 + 2a_2x + 3a_3x^2 + ...$, where the coefficient of $x^n$ is $n*a_n$, representing the total number of ways to select an object of size $n$ and mark one of its elements
• The integral of a generating function $G(x)$ is $\int G(x)dx = a_0x + \frac{a_1}{2}x^2 + \frac{a_2}{3}x^3 + ...$, where the coefficient of $x^{n+1}$ is $\frac{a_n}{n+1}$, representing the number of ways to build an object of size $n+1$ by adding a marked element to an object of size $n$
• The logarithm of a generating function $G(x)$ with $G(0) = 1$ is $\log G(x) = b_1x + \frac{b_2}{2}x^2 + \frac{b_3}{3}x^3 + ...$, where $b_n$ counts the number of connected structures of size $n$ in the combinatorial class associated with $G(x)$ (connected graphs, irreducible polynomials)
• The exponential of a generating function $G(x)$ with $G(0) = 0$ is $\exp G(x) = 1 + G(x) + \frac{G(x)^2}{2!} + \frac{G(x)^3}{3!} + ...$, which can be used to count structures that can be decomposed into connected components (set partitions, forests)
• The Lagrange inversion formula can be used to extract coefficients from generating functions of the form $G(x) = x * \phi(G(x))$, where $\phi$ is a formal power series (Cayley's formula for labeled trees, Ramanujan's continued fractions)

Coefficient extraction from functions

Extraction techniques

• The coefficient $a_n$ in an ordinary generating function $G(x)$ can be extracted using the formula $a_n = [x^n]G(x)$, representing the coefficient of $x^n$ in $G(x)$
• Taylor series expansion can be used to extract coefficients from a generating function by expanding the function around $x=0$ and reading off the coefficients (exponential generating functions, Bell numbers)
• Partial fraction decomposition can be used to split a generating function into simpler terms, making it easier to extract coefficients using geometric series or other known expansions (rational generating functions, Fibonacci numbers)
• The snake oil method (also known as the method of undetermined coefficients) can be used to extract coefficients by assuming a form for the generating function and solving for the unknown coefficients (derangement numbers, Eulerian numbers)
• Lagrange inversion formula can be used to extract coefficients from generating functions of the form $G(x) = x * \phi(G(x))$, where $\phi$ is a formal power series (Catalan numbers, functional graphs)

Applications and examples

• Coefficient extraction techniques can be used to derive closed-form expressions for combinatorial sequences, which can then be used to analyze their asymptotic behavior or establish bijections with other combinatorial objects
• Coefficient extraction can be used to solve problems related to the enumeration of combinatorial structures satisfying certain constraints (permutations avoiding patterns, graphs with specific properties)
• Coefficient extraction can be used to derive identities involving combinatorial sequences by manipulating their generating functions and comparing coefficients (binomial coefficients, Stirling numbers)
• Coefficient extraction can be used to study the distribution of combinatorial statistics by considering the generating functions for the associated probability distributions (Eulerian numbers, q-analogs)
• Coefficient extraction techniques can be combined with other methods, such as combinatorial arguments or bijective proofs, to provide a deeper understanding of combinatorial structures and their properties (lattice paths, Young tableaux)