💁🏽Algebraic Combinatorics Unit 2 – Generating Functions & Recurrences

Key Concepts

• Generating functions encode sequences as coefficients of power series enabling powerful analytic techniques
• Recurrence relations define sequences recursively by expressing later terms as functions of earlier terms
• Ordinary generating functions (OGFs) have coefficients that directly correspond to the terms of a sequence
• Exponential generating functions (EGFs) have coefficients that correspond to the terms of a sequence divided by a factorial
• Solving recurrence relations involves finding closed-form expressions for the nth term of a sequence
• Operations on generating functions (addition, multiplication, differentiation, integration) correspond to operations on the underlying sequences
• Combinatorial identities and properties can often be elegantly proved using generating function manipulations
• Generating functions have applications in enumeration, probability, and analysis of algorithms

Generating Functions Basics

• A generating function is a formal power series where the coefficients encode information about a sequence
• The coefficient of $x^n$ in a generating function $A(x)$ is denoted $[x^n]A(x)$ and corresponds to the nth term of the sequence
• Generating functions can be viewed as a way of packaging the entire sequence into a single mathematical object
• The two main types of generating functions are ordinary generating functions (OGFs) and exponential generating functions (EGFs)
• OGFs are typically used for sequences with simple combinatorial interpretations (e.g., number of ways to select items)
• EGFs are often used for sequences related to labeled structures or where order matters (e.g., permutations)
• Generating functions can be manipulated using algebraic operations to reveal properties of the underlying sequences
• The convergence behavior of a generating function can provide insights into the asymptotic growth of the sequence

Types of Generating Functions

• Ordinary generating functions (OGFs) have the form $\sum_{n=0}^{\infty} a_n x^n$ where $a_n$ is the nth term of the sequence
  • Example: the OGF for the sequence $1, 1, 1, \ldots$ is $\frac{1}{1-x}$
• Exponential generating functions (EGFs) have the form $\sum_{n=0}^{\infty} a_n \frac{x^n}{n!}$ where $a_n$ is the nth term of the sequence
  • Example: the EGF for the sequence $1, 1, 1, \ldots$ is $e^x$
• Bivariate generating functions involve two variables and can encode sequences with two parameters
• Multivariate generating functions generalize to sequences with multiple parameters
• Dirichlet generating functions have the form $\sum_{n=1}^{\infty} \frac{a_n}{n^s}$ and are used in analytic number theory
• Lambert series have the form $\sum_{n=1}^{\infty} \frac{a_n x^n}{1-x^n}$ and have applications in partition theory
• Poisson generating functions are a variant of exponential generating functions used in probability theory

Solving Recurrence Relations

• A recurrence relation defines a sequence recursively by expressing later terms as a function of earlier terms
• Solving a recurrence relation means finding a closed-form expression for the nth term of the sequence
• Generating functions can be used to solve recurrence relations by translating the recurrence into an equation involving generating functions
• The steps for solving a recurrence using generating functions are:
  1. Set up a generating function equation based on the recurrence
  2. Solve the equation for the generating function using algebraic manipulations
  3. Extract the nth term of the sequence from the generating function (e.g., using partial fractions or Taylor series)
• Example: consider the recurrence $a_n = a_{n-1} + a_{n-2}$ with $a_0 = 0$ and $a_1 = 1$ (Fibonacci numbers)
  • The generating function equation is $A(x) = xA(x) + x^2A(x) + x$
  • Solving for $A(x)$ yields $A(x) = \frac{x}{1-x-x^2}$
  • Extracting the nth term (e.g., using Binet's formula) gives the closed-form expression for the Fibonacci numbers
• Other techniques for solving recurrences include the characteristic equation method and the annihilator method

Techniques and Operations

• Addition of generating functions corresponds to adding the underlying sequences term-wise
  • Example: if $A(x)$ generates $a_n$ and $B(x)$ generates $b_n$, then $A(x) + B(x)$ generates $a_n + b_n$
• Multiplication of generating functions corresponds to the convolution of the underlying sequences
  • The convolution of sequences $a_n$ and $b_n$ is defined as $(a * b)_n = \sum_{k=0}^n a_k b_{n-k}$
  • Example: if $A(x)$ generates $a_n$ and $B(x)$ generates $b_n$, then $A(x)B(x)$ generates $(a * b)_n$
• Differentiation of generating functions corresponds to shifting and scaling the underlying sequence
  • For OGFs, $\frac{d}{dx}A(x)$ generates the sequence $na_n$
  • For EGFs, $\frac{d}{dx}A(x)$ generates the sequence $a_{n+1}$
• Integration of generating functions corresponds to the inverse operation of differentiation
• Composition of generating functions corresponds to substitution of sequences
  • Example: if $A(x)$ generates $a_n$ and $B(x)$ generates $b_n$, then $A(B(x))$ generates the sequence $\sum_{k=0}^n a_k b_{n,k}$ where $b_{n,k}$ counts the number of ways to partition a set of size $n$ into $k$ nonempty subsets
• Partial fraction decomposition is a technique for extracting coefficients from rational generating functions
• The Lagrange inversion formula is a powerful tool for extracting coefficients from implicitly defined generating functions

Applications in Combinatorics

• Generating functions are widely used in enumerative combinatorics to count combinatorial objects
• The multiplication principle for generating functions states that if $A(x)$ counts objects of type $A$ and $B(x)$ counts objects of type $B$, then $A(x)B(x)$ counts pairs of objects where the first is of type $A$ and the second is of type $B$
• The composition principle for generating functions states that if $A(x)$ counts objects of type $A$ and $B(x)$ counts objects of type $B$, then $A(B(x))$ counts objects obtained by substituting objects of type $B$ into objects of type $A$
• Generating functions can be used to prove combinatorial identities by showing that two generating functions are equal
  • Example: the identity $\sum_{k=0}^n \binom{n}{k}^2 = \binom{2n}{n}$ can be proved by showing that both sides have the same generating function $\frac{1}{\sqrt{1-4x}}$
• Generating functions can be used to analyze the distribution of combinatorial statistics (e.g., number of inversions in a permutation)
• Generating functions are useful for studying integer partitions, compositions, and other combinatorial structures
• The symbolic method is a powerful framework for deriving generating functions for combinatorial classes based on their structural specifications

Common Patterns and Examples

• The geometric series $\frac{1}{1-x}$ is the OGF for the constant sequence $1, 1, 1, \ldots$
• The exponential series $e^x$ is the EGF for the constant sequence $1, 1, 1, \ldots$
• The binomial series $(1+x)^n$ is the OGF for the sequence of binomial coefficients $\binom{n}{k}$ for fixed $n$
• The Catalan numbers $C_n = \frac{1}{n+1}\binom{2n}{n}$ have the OGF $\frac{1-\sqrt{1-4x}}{2x}$ and count various combinatorial objects (e.g., Dyck paths, binary trees)
• The Fibonacci numbers have the OGF $\frac{x}{1-x-x^2}$ and satisfy the recurrence $F_n = F_{n-1} + F_{n-2}$ with $F_0 = 0$ and $F_1 = 1$
• The Stirling numbers of the first kind $s(n,k)$ have the EGF $\frac{(\log(1+x))^k}{k!}$ and count permutations with $k$ cycles
• The Stirling numbers of the second kind $S(n,k)$ have the EGF $\frac{(e^x-1)^k}{k!}$ and count partitions of a set of size $n$ into $k$ nonempty subsets
• The Bell numbers $B_n$ have the EGF $e^{e^x-1}$ and count the total number of partitions of a set of size $n$

Advanced Topics and Extensions

• Multivariate generating functions can be used to study sequences with multiple parameters
  • Example: the bivariate generating function $\frac{1}{1-x(1+y)}$ encodes the sequence $\binom{n}{k}$ as the coefficient of $x^n y^k$
• Asymptotic analysis of sequences can be performed using the singularity analysis of generating functions
  • The location and nature of the singularities (poles, branch points) of a generating function determine the asymptotic growth of the sequence
  • Example: the Catalan numbers $C_n \sim \frac{4^n}{n^{3/2}\sqrt{\pi}}$ as $n \to \infty$ can be derived from the singularity analysis of their generating function
• Generating functions can be interpreted as analytic objects in complex analysis
  • The study of generating functions as complex functions leads to powerful results in analytic combinatorics
  • Example: the Hadamard product of generating functions corresponds to the termwise product of the underlying sequences and can be used to prove asymptotic formulas
• Umbral calculus is a symbolic approach to generating functions that treats the variable $x$ as a formal object with special properties
  • Umbral operators and identities provide a concise notation for manipulating generating functions
  • Example: the umbral composition formula $(f(t)g(t))^k = f(t)^k g(f(t)t)^k$ relates the composition of generating functions to their powers
• Generating functions have connections to other areas of mathematics, such as number theory, probability theory, and mathematical physics
  • Example: the partition generating function $\prod_{n=1}^{\infty} \frac{1}{1-x^n}$ is related to the Dedekind eta function in modular form theory
• Automated methods for finding and proving generating function identities have been developed using computer algebra systems
  • Example: the Wilf-Zeilberger method uses creative telescoping to prove hypergeometric identities involving generating functions