🧮Combinatorics Unit 7 – Recurrence Relations

Key Concepts and Definitions

• Recurrence relations define sequences 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 term
  • Example: $a_n = 2a_{n-1} + 3a_{n-2}$
• Non-homogeneous recurrence relations include an additional term that is not a multiple of the preceding terms
  • 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
• Generating functions represent sequences as coefficients of a power series and can be used to solve recurrence relations

Types of Recurrence Relations

• Linear recurrence relations express each term as a linear combination of the preceding terms
  • Example: $a_n = 3a_{n-1} - 2a_{n-2}$
• Non-linear recurrence relations involve non-linear functions of the preceding 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 change with $n$
• Variable-coefficient recurrence relations have coefficients that vary with $n$
  • Example: $a_n = na_{n-1} + (n-1)a_{n-2}$

Solving Linear Recurrence Relations

• Solve homogeneous linear recurrence relations by finding the characteristic equation
  • Substitute $a_n = r^n$ into the recurrence relation and solve for $r$
  • The roots of the characteristic equation determine the form of the general solution
• For distinct roots, the general solution is a linear combination of the roots raised to the power $n$
  • Example: If the roots are $r_1$ and $r_2$, the general solution is $a_n = c_1r_1^n + c_2r_2^n$
• For repeated roots, the general solution includes polynomial factors multiplied by the repeated root raised to the power $n$
• Use the initial conditions to determine the values of the constants in the general solution
• Solve non-homogeneous linear recurrence relations by finding a particular solution and adding it to the general solution of the homogeneous part
  • The method of undetermined coefficients can be used to find a particular solution

Generating Functions and Recurrences

• Ordinary generating functions (OGFs) represent sequences as power series
  • The coefficient of $x^n$ in the generating function corresponds to the $n$-th term of the sequence
• Exponential generating functions (EGFs) are similar to OGFs but include a factor of $\frac{1}{n!}$ in each term
• Generating functions can be used to solve recurrence relations by translating the recurrence into an equation involving the generating function
  • Multiply both sides of the recurrence by $x^n$ and sum over all $n$ to obtain an equation for the generating function
• Partial fraction decomposition can be used to find the coefficients of the generating function and thus the terms of the sequence
• Generating functions can also be used to find closed-form expressions for the terms of a sequence

Applications in Combinatorics

• Recurrence relations can model the growth of populations, such as the Fibonacci sequence for rabbit populations
• Counting problems can often be solved using recurrence relations
  • Example: The number of ways to climb a staircase with $n$ steps, taking either one or two steps at a time
• Generating functions can be used to count the number of ways to partition a set or integer
  • The partition function $p(n)$ counts the number of ways to partition an integer $n$ into smaller parts
• Recurrence relations and generating functions can be used to analyze the time complexity of recursive algorithms
  • The Merge Sort algorithm has a recurrence relation of $T(n) = 2T(\frac{n}{2}) + O(n)$

Problem-Solving Strategies

• Identify the type of recurrence relation (linear, non-linear, homogeneous, non-homogeneous)
• For linear recurrence relations, find the characteristic equation and solve for the roots
• Use the initial conditions to determine the constants in the general solution
• For non-homogeneous recurrence relations, find a particular solution and add it to the general solution of the homogeneous part
• Consider using generating functions to solve the recurrence relation or find a closed-form expression
• Break down complex recurrence relations into simpler subproblems
• Look for patterns or similarities to known recurrence relations or sequences

Common Pitfalls and Mistakes

• Forgetting to include the initial conditions when solving a recurrence relation
• Incorrectly setting up the characteristic equation or solving for the roots
• Neglecting to find a particular solution for non-homogeneous recurrence relations
• Misinterpreting the meaning of the coefficients in a generating function
• Attempting to solve a non-linear recurrence relation using methods meant for linear recurrences
• Failing to recognize when a recurrence relation can be simplified or transformed into a more manageable form
• Overlooking the importance of boundary conditions or special cases in a recurrence relation

Advanced Topics and Extensions

• Matrix methods for solving systems of linear recurrence relations
• Recurrence relations with more than one variable, such as $a_{m,n}$
• Asymptotic analysis of recurrence relations using tools like the Master Theorem or the Akra-Bazzi method
• Generating functions for multivariate sequences or sequences with additional constraints
• Continued fractions and their relationship to recurrence relations and generating functions
• Orthogonal polynomials and their recurrence relations, such as the Chebyshev or Legendre polynomials
• Recurrence relations in the context of dynamic programming and optimization problems