A Brief Journey in Discrete Mathematics by Randolph Nelson
Contents of Brief Journey in Discrete Mathematics
- Introduction
- Let Me Count the Ways
- Permutations: With and Without Replacement
- Dearrangements
- Combinations: Without Replacement
- Binomial Identities
- Combinations with Replacement
- Binomial-R Identities
- Polynomial Solutions to Combinatorial
- Problems
- Transforms and Identities
- Syntax Precedes Semantics
- Stirling Numbers of the First Kind
- Stirling Numbers of the Second Kind
- The Stirling Transform and Inverse
- Combinatorial Interpretation
- Fearful Symmetry
- Symmetric Functions
- Simple Polynomials
- The Quadratic Equation
- Equation of the Minimum Distance Line
- The Pythagorean Theorem
- Cubic Polynomials
- Elementary Symmetric Polynomials
- Newton–Girard Formula
- Identities and Combinatorial Coefficients
- Inclusion–Exclusion
- Fundamental Theorem of Symmetric Polynomials
- Galois’ Theorem and Numerical Solutions
- All That Glitters Is Not Gold
- The Golden Ratio
- Fibonacci Numbers
- A Closed Form Solution
- An Alternate Derivation
- Generalized Fibonacci Numbers
- k-Bonacci Numbers
- Generalization of the Fibonacci Recurrence
- Heads I Win, Tails You Lose
- The Mathematical Model
- Games That End Even
- Catalan Numbers
- Non-intuitive Results
- The Correct Insight
- Particular Sequences
- Conclusions
- Sums of the Powers of Successive Integers
- A General Equation
- Iterative Approach
- Triangular Numbers
- Cauchy’s Theorem
- As Simple as + =
- Modular Arithmetic
- Fermat’s Little Theorem
- Lagrange’s Theorem
- Wilson’s Theorem
- Cryptography
- Hidden in Plain Sight
- Properties of Prime Numbers
- Properties of Integer Divisors
- The Prime Counting Function
- There Is Always a Prime Between n and n
- The Prime Number Theorem with a
- Controversy
- Running Off the Page
- Simple Continued Fractions
- Periodic Simple Continued Fractions
- Summary of Results
- General Method to Create a Continued Fraction
- Integer Quadratics and Quadratic Surds
- Approximations Using Continued Fractions
- Best Approximations
- Lagrange’s Theorem and Historical Review
- A Tools of the Trade
- A Recurrence Relationships
- A Adding Zero to an Equation
- A Induction
- A Contradiction
- A Order of Summations
- B Notation and Identities Derived in the Book
