Day 1 (1/6/2020):

• Discussion of the syllabus/administrative business

• Discussed the 4 basic principles of counting (addition, multiplication, subtraction, division), ordered enumeration v.s. unordered enumeration, counting with and without repetition, worked some introductory examples

• Textbook reference: Section 2.1 of Brualdi (roughly)

Day 2 (1/8/2020):

• Introduced permutations (ordered enumeration)

• Introduced combinations(unordered enumeration)

• Textbook reference: Sections 2.2-2.3 Brualdi

Day 3 (1/10/2020):

• Introduced mutliset permutations, nonattacking rook configurations

• Introduced multiset combinations

• Textbook Reference: Sections 2.4-2.5 Brualdi

Day 4 (1/13/2020):

• Further discussion of multiset combinations; counting nonnegative integer solutions to a linear equations

• Finite Probability; Poker hands

• Textbook reference: Sections 2.5-2.6 Brualdi

Day 5 (1/15/2020):

• Substitute Lecturer: Dr. Eric Katz

• Pigeonhole Principle (simple and strong forms)

• Textbook Reference: Sections 3.1-3.2 Brualdi

Day 6 (1/22/2020):

• Introductory Ramsey Theory: using pigeonhole principle to for monochromatic triangles, existence of Ramsey numbers, Ramsey number inequality

• Textbook Reference: Section 3.3 Brualdi

Day 7 (1/24/2020):

• Binomial Coefficients Part 1

• Combinatorial Interpretations: subsets, bit strings, lattice paths

• Combinatorial Proofs

• Binomial Theorem

• Textbook Reference: parts of sections 5.1 and 5.2 Brauldi; section 1.11 Sagan

Day 8 (1/27/2020):

• Binomial Coefficients Part 2

• More combinatorial proof and proofs with binomial theorem

• Textbook ref: sec. 5.3 Brualdi

Day 9 (1/29/2020):

• Binomial Coefficients Part 3

• Multinomial Coefficients and the Multinomial Theorem

• Newton's Binomial Theorem, approximating square roots

• Textbook reference: sec. 5.4-5.5 Brualdi

Day 10 (1/31/2020):

• Finished discussion on approximating square roots

• Introduced Posets

• Dilworths Thm.

• Textbook ref: sec 5.6 (and 4.5 to a degree) Brualdi

Day 11 (2/3/2020):

• Finished discussion of Dilworth's Thm

• Begin Section of Sieve Methods

• Introduce Principle of Inclusion-Exclusion (PIE)

Day 12 (2/5/2020):

• PIE applied to multiset permutations

• PIE applied to Derangements

Exam Day (2/7/2020)

Day 13 (2/10/2020):

• Recap of derangements (probability of a derangement from a random permutation)

• Generalizing derangements--forbidden positions in a permutation

Day 14 (2/14/2020):

• Relative forbidden positions in a permutation (PIE)

• Introduced Sign-Reversing Involutions (See Sec. 2.2 of Counting the Art of Combinatorics)

Day 15 (2/17/2020):

• Substitute Lecturer: Dr. Hans Pashall

• Catalan Numbers (Combinatorial definition, proof of closed form, convolution recurrence, other Catalan objects)

• Lecture Notes for reference

Day 16 (2/19/2020):

• More Sign-Reversing Involutions

• Set Partitions and Stirling Numbers (of the 2nd kind)

Day 17 (2/21/2020):

• A sign-reversing involution using Stirling numbers/set partitions

• Introduction to Ordinary Generating Functions (OGFs)

• Using OGFs for a closed form of the Fibonacci Numbers

Day 18 (2/24/2020):

• Another example of using OGFs to find closed forms for recurrence relations

• Introduced compositions

• Proved that compositions with 1's and 2's are enumerated by Fibonacci numbers

• Briefly discussed how to construct an OGF for counting compositions with various restrictions.

Day 19 (2/26/2020):

• An example of an OGF for counting compositions containing only certain parts with various colors or weights

• An example of an OGF for weighted compositions with no restriction on parts

• Introduction to Exponential Generating Functions (EGFs)

Day 20 (2/28/2020):

• EGF example for a multiset permutation with various restrictions on when elements can appear

• EGF example for coloring squares of a 1 x n chessboard with various restrictions

• Return to OGFs: Began computing the Catalan OGF

Day 21 (3/2/2020):

• Finished discussion of the Catalan OGF

• Concluded (direct) discussion of generating functions by stating a Theorem: A sequence has a rational ordinary generating function if and only if the sequence can be expressed as a linear recurrence relation with constant coefficients.

• Began discussing q-analogues:

  • q-integers

  • q-factorials: Both the ``naive" description from q-integers, as well as the combinatorial description from inversions in permutations. Proved these are equivalent.

Day 22 (3/4/2020):

• Discussed the 4 equivalent formulations of the q-binomial coefficient

• Stated the q-binomial theorem

Day 23 (3/6/2020):