Combinatorics

Week 1 | Week 2

Tuesday, January 18

Class notes

• Permutations
• Factorials
• Binomial coefficients
• Lattice paths

Thursday, January 20

Class notes

• Pascal’s triangle recurrence
• The binomial theorem
• Summing 1 through n
• Bars & stars for counting monomials

Suggested reading

I particularly recommend these expositions, from easy to advanced:

• 2015 Beeler - How to count
  • 1 Preliminaries
  • 2 Basic Counting
  • 3 The Binomial Coefficient
• 2011 Camina, Lewis - An introduction to enumeration
  • 1.1 Bijections,Permutations and Sequences
• 2007 Aigner - A Course in Enumeration
  • 1.1 Elementary Counting Principles
  • 1.2 Subsets and Binomial Coefficients

You may prefer these expositions:

• 1983 Pólya, Tarjan, Woods - Notes on Elementary Combinatorics
  • 1. Combinations and Permutations
• 2001 Martin - Counting: The art of enumerative combinatorics
  • Chapter 1. Elementary Enumeration
• 2003 Lovasz, Pelikan, Vesztergombi - Discrete mathematics
  • 1 Let’s Count!
• 2015 Petersen - Eulerian numbers
  • 1.1 Binomial coefficients
• 2016 Mariconda, Tonolo - Discrete calculus
  • 2.1 Sequences and Collections Having Distinct Elements

Along the way, try exercises that look interesting to you.