New definitions are in bold and key topics covered are in a bulleted list.
This schedule is approximate and subject to change!

1/28: Sections 1.0, 1.1, 1.2, and 1.5 (Notes pages 1-8)

• What is combinatorics?
• Example: Domino Tilings
• Example: Slicing a cube
• Example: Domino Tiling Fault Lines
• Example: Orthogonal Latin Squares

2/2: Sections 1.7 and 3.1 (Notes pages 9-13)

• Example: Let's play Nim!
• The addition and multiplication principles.

2/4: Sections 3.2 and 3.3 (Notes pages 14-19)

• The subtraction and division principles.
• Differences between permutations and combinations
• r-Permutations of sets with n elements.
• r-Combinations of sets with n elements.
• Examples involving permutations and combinations.
• Circular permutations.

2/9: Sections 3.4 and 3.5 (Notes pages 20-23)

• Multisets.
• Physical representation of permutations and combinations of sets and multisets.
• Permutations and combinations of multisets
• Examples of permutations of multisets

2/11: Homework 3 discussion

2/16: No class, Presidents Day

2/18: Sections 3.5, 3.3, and 5.3 (Notes pages 24-30)

• Non-attacking rooks
• Examples of combinations of multisets
• Combinatorial proofs and combinatorial interpretations

2/23: Section 5.3 (Notes pages 31-33)

• Worksheet on combinatorial proofs
• Bijections

2/25: Homework 5 discussion

3/2: Snow Day

3/4: Sections 5.1, 5.2, and 5.3 (Notes pages 34-39)

• Pascal's formula
• Pascal's triangle
• The binomial theorem and related formulas
• Lattice paths

3/9: Sections 5.3, 5.6, and 5.5 (Notes pages 40-44)

• Extensions of binomial coefficients
• Newton's binomial theorem
• Multinomial Theorem
• Multinomial Coefficients

3/11: Question and Answer Session

3/16: Exam 1

3/18: Sections 6.1, 6.2, and 6.3 (Notes pages 44-49)

• Venn Diagrams
• Inclusion/Exclusion: Determining how many elements do satisfy properties.
• Inclusion/Exclusion: Determining how many elements don't satisfy properties.
• Groupwork on inclusion/exclusion.
• Derangements

3/23: Sections 6.3, 6.4, and 7.1 (Notes pages 49-53)

• The technique of partial fractions
• Combinatorial proofs involving derangements.
• Permutations with forbidden positions
• Rook interpretation with forbidden positions
• Integer Sequences

3/25: Sections 7.1, 7.2 (Notes pages 53-58)

• Fibonacci numbers
• Square-domino interpretation of Fibonacci numbers
• Introduction to Generating Functions

3/30: Section 7.4 (Notes pages 59-63)

• Generating functions
• Generating function manipulation
• Worksheet: determining a generating function from a sequence

4/1: Homework 9 discussion

4/6: Sections 7.5

• Worksheet: determining a sequence from its generating function.
• Application: Relabeling dice to give the same sum distribution
• Counting combinations of fruit with restrictions
• Using generating functions to solve recurrences

4/8: No class, Spring Break

4/13: No class, Spring Break

4/15: No class, Spring Break

4/20: Section 7.5 and 8.2

• Using generating functions to solve recurrences
• Partitions of a set and of a number
• Partitions of a set
• Distinguishable/Indistinguishable boxes/objects
• Combinatorial interpretation of Stirling numbers of the second kind
• A recurrence for Stirling numbers of the second kind

4/22: Sections 8.2 and 8.3

• A formula for Stirling numbers of the second kind
• Bell numbers and a recurrence they satisfy.
• Partitions of a number
• Ferrers diagram of a partition
• Conjugate partitions
• Partition numbers and generating functions
• Bijective proofs involving partition numbers
• diagrams, tableaux, semi-standard tableaux, standard tableaux
• Standard Young tableaux
• hook length formula
• Also: hook length of a cell

4/27: Sections 7.6, 8.1

• Catalan numbers
• Combinatorial interpretations of Catalan numbers
• Bijections between the various interpretations
• Dyck paths
• Catalan recurrences

4/29: Sections 7.6, 8.1

• Catalan numbers
• Combinatorial interpretations of Catalan numbers
• Bijections between the various interpretations
• Dyck paths
• Catalan recurrences

5/4: Sections 7.6, 8.1

• Generating function for Catalan numbers
• Proof of the formula for the Catalan numbers
• Combinatorial interpretation of the Catalan generating function.

5/6: Question and Answer Session

5/11: No class, poster work day

5/13: Poster Presentations

Monday, 5/18: Exam 2, 4-6pm in KY 427