🔢Enumerative Combinatorics Unit 5 – Recurrence Relations

Key Concepts and Definitions

• Recurrence relations define a sequence recursively by expressing each term as a function of the preceding terms
• Initial conditions specify the values of the first few terms in the sequence
• Homogeneous recurrence relations have a constant coefficient and no additional terms
  • Example: $a_n = 2a_{n-1} + 3a_{n-2}$
• Non-homogeneous recurrence relations include additional terms or functions
  • Example: $a_n = 2a_{n-1} + 3a_{n-2} + n$
• Characteristic equation is derived from the recurrence relation and used to find the general solution
• Closed-form solution expresses the nth term of the sequence explicitly as a function of n

Types of Recurrence Relations

• Linear recurrence relations express each term as a linear combination of previous terms
  • Example: $a_n = 3a_{n-1} - 2a_{n-2}$
• Non-linear recurrence relations involve higher powers or products of previous terms
  • Example: $a_n = a_{n-1}^2 + a_{n-2}$
• First-order recurrence relations depend only on the immediately preceding term
  • Example: $a_n = 2a_{n-1} + 1$
• Higher-order recurrence relations depend on multiple preceding terms
• Constant-coefficient recurrence relations have coefficients that do not depend on n
• Variable-coefficient recurrence relations have coefficients that are functions of n
  • Example: $a_n = na_{n-1} + (n-1)a_{n-2}$

Solving Linear Recurrence Relations

• Homogeneous linear recurrence relations with constant coefficients can be solved using the characteristic equation method
  1. Write the characteristic equation by replacing $a_n$ with $x^n$ and equating to zero
  2. Solve the characteristic equation to find the roots
  3. Express the general solution as a linear combination of the roots raised to the power of n
• Non-homogeneous linear recurrence relations can be solved using the method of undetermined coefficients or variation of parameters
  • Particular solution is added to the homogeneous solution to obtain the general solution
• Initial conditions are used to determine the specific values of the constants in the general solution
• Generating functions can be used to solve linear recurrence relations by transforming them into algebraic equations

Generating Functions and Recurrences

• Generating functions encode the terms of a sequence as coefficients of a power series
  • Example: $A(x) = a_0 + a_1x + a_2x^2 + \ldots$
• Ordinary generating functions (OGFs) are used for sequences indexed by non-negative integers
• Exponential generating functions (EGFs) are used for sequences with a combinatorial interpretation
  • Example: $A(x) = a_0 + a_1\frac{x}{1!} + a_2\frac{x^2}{2!} + \ldots$
• Recurrence relations can be transformed into equations involving generating functions
  • Multiply both sides of the recurrence by $x^n$ and sum over all n
• Partial fraction decomposition is used to extract coefficients from rational generating functions
• Generating functions can be used to solve problems involving counting, probability, and combinatorial identities

Applications in Combinatorics

• Recurrence relations arise naturally in many combinatorial problems
  • Counting the number of ways to arrange objects with certain constraints
  • Enumerating the number of paths in a graph or grid
• Catalan numbers satisfy the recurrence relation $C_n = \sum_{i=0}^{n-1} C_iC_{n-1-i}$ and count various combinatorial objects
  • Examples: balanced parentheses, binary trees, polygon triangulations
• Fibonacci numbers follow the recurrence $F_n = F_{n-1} + F_{n-2}$ and appear in many combinatorial settings
  • Examples: rabbit population growth, tiling a 2xn rectangle with dominoes
• Stirling numbers of the second kind $S(n, k)$ count the number of ways to partition a set of n elements into k non-empty subsets
  • Satisfy the recurrence $S(n, k) = S(n-1, k-1) + kS(n-1, k)$
• Derangements are permutations with no fixed points and satisfy the recurrence $D_n = (n-1)(D_{n-1} + D_{n-2})$

Advanced Techniques and Special Cases

• Generating function techniques can be extended to multivariate generating functions for sequences with multiple indices
• Lagrange inversion formula allows the inversion of power series and can be used to solve certain recurrence relations
• Asymptotic analysis provides estimates for the growth rate of sequences defined by recurrence relations
  • Examples: using the Master Theorem for divide-and-conquer recurrences, applying the Akra-Bazzi method for more general recurrences
• Recurrence relations with non-constant coefficients often require specialized techniques
  • Examples: using the Wilf-Zeilberger method for hypergeometric recurrences, applying the Birkhoff-Trjitzinsky method for linear recurrences with rational coefficients
• Matrix methods can be used to solve systems of linear recurrence relations
  • Diagonalization of the companion matrix leads to closed-form solutions

Problem-Solving Strategies

• Identify the recurrence relation and initial conditions that describe the problem
• Determine the type of recurrence relation (linear, non-linear, homogeneous, non-homogeneous)
• Choose an appropriate solution method based on the type of recurrence and the desired form of the solution
  • Examples: characteristic equation method for homogeneous linear recurrences, generating functions for problems involving counting or probability
• Verify that the solution satisfies the recurrence relation and initial conditions
• Analyze the asymptotic behavior of the solution if necessary
  • Examples: using Big-O notation to describe growth rates, applying asymptotic techniques for recurrences
• Consider alternative approaches or simplifications if the initial approach proves difficult
  • Examples: transforming the recurrence into a more manageable form, exploiting symmetries or combinatorial identities

Connections to Other Topics

• Recurrence relations are closely related to difference equations, which are discrete analogues of differential equations
• Generating functions have applications in various areas of mathematics, including formal power series, analytic combinatorics, and number theory
• Recurrence relations appear in the analysis of algorithms, particularly in the study of divide-and-conquer and dynamic programming algorithms
  • Examples: merge sort, quicksort, Fibonacci sequence computation
• Combinatorial identities often have proofs based on recurrence relations or generating functions
  • Examples: binomial theorem, Vandermonde's identity, Catalan number identities
• Recurrence relations are used in the study of discrete dynamical systems and population models
  • Examples: Fibonacci rabbit population model, logistic growth model
• Generating functions and recurrence relations have applications in probability theory and statistics
  • Examples: probability generating functions, moment generating functions, Markov chains