8/29: Sections 1.2 and 1.5

• Example: Domino Tiling Fault Lines
• Example: Orthogonal Latin Squares

8/30: Section 1.7

• Discussion of homework
• Example: Let's play Nim!

8/31: Section 2.1

• The (weak) pigeonhole principle

          Week 2 -- Principles

9/3: No class, Labor Day

9/5: Section 2.2

• The strong pigeonhole principle

9/6: Section 3.1

• The addition and multiplication principles.

9/7: Section 2.4

• Discussion of homework 2B

          Week 3 -- More Counting

9/10: Sections 3.1 and 3.2

• The subtraction and division principles.
• Reading report question discussion

9/12: More discussion

• Discussion of Homework 2A.
• Chinese remainder theorem. (pp. 30-31)

9/13: No class

9/14: No class

          Week 4 -- Permutations and Combinations

9/17: Sections 3.2 and 3.3

• Differences between permutations and combinations
• r-Permutations of sets with n elements.
• r-Combinations of sets with n elements.
• Examples involving permutations and combinations.

9/19: Sections 3.4 and 3.5

• Multisets.
• Physical representation of permutations and combinations of sets and multisets.
• Mississippi

9/20: Sections 3.1-3.5

• Homework 4B discussion
• Reading report discussion

9/21: Sections 3.3, 3.4, and 3.5

• What is a combinatorial proof?
• Examples of permutations and combinations of multisets

          Week 5 -- Binomial Coefficients

9/24: Section 3.5

• r-combinations of multisets
• Pascal's formula and combinatorial proofs.

9/26: Section 5.1

• Pascal's triangle
• The binomial theorem

9/27: Discussion

• Homework 5B discussion
• Reading Report discussion

9/28: Discussion

• Homework 5B discussion
• Reading Report discussion

          Week 6 -- Identititititities

10/1: Section 5.2

• Worksheet on Combinatorial Proofs

10/3: Question and Answer Session

10/4: Exam 2

10/5: Sections 5.3 and 5.6

• Extensions of binomial coefficients
• Pascal's binomial theorem

          Week 7 -- Inclusion/Exclusion

10/8: Sections 5.5 and 6.1

• Multinomial Theorem
• Multinomial Coefficients
• Venn Diagrams
• Inclusion/Exclusion: Determining how many elements do satisfy properties.

10/10: Sections 6.1, 6.2

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

10/11: Discussion

• Homework 7B discussion
• Convolution: Multiplication of two infinite series.

10/12: Section 6.3

• Derangements
• Combinatorial proofs involving derangements.

          Week 8 -- Sequences and Recurrence Relations

10/15: Section 6.4

• Permutations with forbidden positions
• Rook interpretation with forbidden positions

10/17: Section 7.1

• Integer Sequences
• Fibonacci numbers
• Square-domino interpretation of Fibonacci numbers

10/18: Discussion

• Homework 8B discussion

10/19: Sections 7.1, 7.2

• Binet's formula for f_n
• General solution to a homogeneous linear recurrence relation with constant coefficients

          Week 9 -- Generating Functions

10/22: Section 7.4

• Generating functions
• Application: Relabeling dice to give the same sum distribution

10/24: Section 7.4

• Generating function manipulation
• Counting combinations of fruit with restrictions
• Worksheet: determining a generating function from a sequence

10/25: Discussion

• Homework 9B discussion

10/26: Sections 7.4, 7.5

• Reading report discussion
• Using generating functions to solve recurrences
• Worksheet: determining a sequence from a generating function

          Week 10 -- Miscellaneous

10/29: Section 7.5

• Using generating functions to solve recurrences
• Worksheet: determining a sequence from a generating function

10/31: Question and Answer Session

11/1: Exam 2

11/2: Sections 7.6, 8.1

• Catalan numbers
• Combinatorial interpretations of Catalan numbers
• Bijections between the various interpretations

          Week 11 -- Catalan Numbers and Dyck Paths

11/5: Section 8.1

• Completion of the bijections
• Dyck paths
• Catalan recurrences
• Generating function for Catalan numbers
• Partitions of a set and of a number

11/7: Section 8.1

• Proof of the formula for the Catalan numbers
• Combinatorial interpretation of the Catalan generating function.
• Review of Exam 2

11/8: Discussion

11/9: Section 8.2

• Review of Exam 2
• Partitions of a set
• Distinguishable/Indistinguishable boxes/objects
• Combinatorial interpretation of Stirling numbers of the second kind

          Week 12 -- Partitions

11/12: Section 8.2

• A recurrence for Stirling numbers of the second kind
• A formula for Stirling numbers of the second kind
• Bell numbers and a recurrence they satisfy.

11/14: Section 8.3

• Partitions of a number
• Ferrers diagram of a partition
• Conjugate partitions
• Partition numbers and generating functions

11/15: Discussion

• Also: hook length of a cell

11/16: Section 8.3

• Bijective proofs involving partition numbers
• diagrams, tableaux, semi-standard tableaux, standard tableaux
• Standard Young tableaux
• hook length formula

          Week 13 -- Partitions

11/19: Section 8.5

• Schröder paths
• Large Schröder numbers
• Generating function, recurrence for the large Schröder numbers

11/21: No class

11/22: No class

11/23: No class

          Week 14 -- Schröder Numbers and Schröder Paths

11/26: Section 8.5

• Small Schröder numbers
• Bracketings, dissections, and a bijection
• Generating function for the small Schröder numbers
• Relationship between the large and small Schröder numbers

11/28: Section 8.5

• Recurrence for the small Schröder numbers
• Preview of Graph Theory next semester.

11/29: Discussion

11/30: In-class poster work day

          Week 15 -- Miscellaneous

12/3: Voting Theory Job Talk

12/5: Second Job Talk

12/6: Poster Presentations

12/7: Question and Answer Session

          Finals Week

Monday, 12/10: Exam 3, 11am-1pm in AAG 007