I've posted the notes and topics for each day and what is expected of you in and out of class. This schedule is approximate and subject to change! Check back here often.

• Notes from Section 1.1 (Notes pages 0–19) '
• Syllabus discussion.
• Simple Counting
• Multiplication Principle
• Examples

• Email me at chanusa@qc.cuny.edu with the following three or four things: (1) Your name, (2) Your class (Math 636) (3) the email address where you are best contacted (4) Which math classes you've taken in the past year and are taking this semester. If you are an undergraduate also send me (5) your graduation year.
• Thoroughly read all pages of the course webpage. This should answer all the questions that you may have about the class.
• Now take the Syllabus Quiz. Feel free to refer back to the course webpage for help. Retake the quiz as many times as necessary to earn a score of 100%.
• Fill out this Doodle to let me know when you are available for office hours this semester. I will try to choose some times that work for everyone.
• (Download / Print out) the notes for Wednesday's class (below)
• Background reading: Combinatorics: A Guided Tour, Section 1.1
• Prepare to answer the following thought questions in class.
  • Question 1. Answer Question 1.1.2ab (This means parts (a) and (b) in Problem 2 of Section 1.1). Then: (c) Assuming the coin is fair, what is the probability that a sequence of 20 flips has exactly 10 heads and 10 tails (in any order)?
  • Question 2. How many orderings are there for a deck of 52 cards if all the cards of the same suit are together?

• Notes from Sections 1.2 & 2.2 (Notes pages 20–28) '

• In the week between August 29 and September 5, meet for one hour outside of class with at least one other classmate. Your goal is to get together to talk about combinatorics, working together to answer the thought questions.
• Write one paragraph about something that you learned or experienced that you would not have if you had worked alone. Also write down the name of the person you met with and when you met! I will collect this paragraph Wednesday and it will count toward your class participation grade.
• Background reading: Combinatorics: A Guided Tour, Sections 1.1 and 1.2
• Prepare to answer the following thought questions in class.
  • Question 3. (a) Answer Question 1.1.5. Then: (b) Among all ways to choose fifteen coins, what is the smallest amount of money that occurs in at least two different ways? (For example, 15 dimes equals $1.50.)
  • Question 4. (a) How many subsets of [30] contain no prime numbers? (b) How many subsets of [30] have size 15 and no numbers larger than 20? (c) How many multisubsets of [30] of size 13 have smallest element 6 and largest element 17?
  • Question 5. Answer Question 1.2.9.

• Continuation of previous set of notes.
• Counting pitfalls
• Overcounting, Counting the complement
• Pascal's triangle and the binomial theorem

• Background reading: Combinatorics: A Guided Tour, Sections 1.1 and 1.2
• Prepare to answer the following thought questions in class.
  • Question 6. Answer each of the following EIGHT jeopardy questions by giving a "real world" situation that could be counted by the given quantity. Exercises 1.2.1abcd, 2.1.1c, 2.2.1acd
  • Question 7. In chess, a rook is a piece that can move only vertically and horizontally. Therefore, two rooks attack each other if they are placed in the same row or in the same column. A non-attacking configuration of rooks consists of placing some number of rooks on a chessboard so that no pair of rooks attack each other. Determine the number of non-attacking configurations of five indistinguishable rooks on an 8x8 chessboard.
  • Question 8. Exercise 2.2.8

• Notes from Section 1.3 (Notes pages 29–35) '
• Bijections
• Bijection Worksheet

• Background reading: Combinatorics: A Guided Tour, Section 1.3
• Prepare to share your thoughts about the exploration discussed here.
  • Exploration Question: Let a_n be the maximal number of pieces into which you can cut a circle using n lines. Determine by hand the first few values of a_n. Use the Online Encyclopedia of Integer Sequences to determine what the formula is for a_n as a function of n.
     
    Once you have found the sequence, there are links right after the first few terms of the sequence. You should look at the graph of the sequence and listen to the sequence. On your homework, write down the 42nd term of the sequence.
     
    Then, via the WebCam link at the bottom of the page, look through a few sequences and write down a sequence that looks interesting (its sequence number, its description, and a few first terms) and last, write one to two paragraphs about why you thought it was interesting. We'll share that with our classmates in class.

• Bijections
• Bijection Worksheet

• Complete the Bijection Worksheet we started in class.
• Prepare to answer the following thought questions in class.
  • Question 9. Suppose k≤n.
    1. How many functions are there from [k] to [n]?
    2. How many bijections are there from [k] to [n]?
    3. How many one-to-one functions are there from [k] to [n]?
    4. How many onto functions from [k] to [n] are not one-to-one?
  • Question 10. Create and prove a bijection between two-member subsets of {1, 2,..., n, n+1} and all possibilities of placing one pair of parentheses in a string of n letters. For example, when n=3, we see that there are six two-member subsets of {1,2,3,4}:

    {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, and {3,4}.

    and there are six ways to place one pair of parentheses in the word abc:

    (a)bc, a(b)c, ab(c), (ab)c, a(bc), and (abc).

    Notice that "a()bc" is not valid—there are no letters between the parentheses.
• The standards that will be assessed on the first assessment on Wednesday, September 26 are now posted. Take a few minutes to read through them.

• Homework Discussion
• Bijections
• Notes from Section 1.4 (Notes pages 36–42) '
• Equivalence classes
• Counting using symmetry

• Prepare for Assessment 1.

• Assessment 1 on Standards 1–4.
• Continuation of notes on Section 1.4
• Equivalence classes
• Counting using symmetry

• Background reading: Combinatorics: A Guided Tour, Section 1.4
• Send me an email by Sunday evening if you want to reassess a standard in office hours.

• Counting using symmetry
• Notes from Sections 2.1, 2.2, and 4.2 (Notes pages 43–51) '
• Combinatorial proofs

• Prepare to answer the following questions in class.
  • Question 11. (a) Exercise 1.4.15
    (b) Write a paragraph explaining why we can not use the equivalence principle to count the number of different necklaces where two of the n beads are indistinguishable (the same color, for example).
  • Question 12. Exercise 1.4.13.

• Homework Discussion
• Tiling interpretation of Fibonacci numbers
• Notes from Sections 2.1 through 2.4 (Notes pages 52–61) '
• Counting distributions
• The sixteenfold way
• Worksheet on Combinatorial proofs

• Prepare to answer the following questions in class.
  • Question 13. Use the square-domino interpretation of the Fibonacci numbers to give a combinatorial proof that f_2n=1+Σ_1≤i≤n f_2i-1.
  • Question 14. Give a combinatorial proof of the identity in Exercise 2.2.4f.
  • Question 15. Understand and explain to the class the proof of Theorem 2.1.2. The proof relies on the argument given in Combinatorial Proof #2 on page 55 and the solution to Question 62.

• Homework Discussion
• Worksheet Discussion

• Prepare to answer the following questions in class.
  • Question 16. Exercise 2.1.6
  • Question 17. Figure out the answer to each of the following parts. Prove at least one of them using a bijection.
    1. How many set partitions of [n] into two blocks are there? (Definition of block on p. 35)
    2. How many set partitions of [n] into (n-1) blocks are there?
    3. How many set partitions of [n] into (n-2) blocks are there?

• Stirling numbers, Bell numbers
• Integer partitions

• Prepare for Assessment 2.
• Prepare to answer the following questions in class.
  • Question 18. Exercise 2.3.4
  • Question 19. Exercise 2.4.1
  • Question 20. Exercise 2.4.11

• Discussion of homework questions.
• Assessment 2 on Standards 5–7.
• Notes from Section 3.1 and pp. 165–166 (Notes pages 62–73) '
• Principle of Inclusion/Exclusion
• Derangements
• Formulas for Stirling numbers and Bell numbers

• Project topic due today.
• Prepare to answer the following questions in class.
  • Question 21. Exercise 3.1.4(a) Hint: Define A₁ to be the set of 13-card hands that have no spades. (Or call it A_♠!)
  • Question 22. Exercise 3.1.17

• Homework Discussion
• Notes from Sections 3.3 and 3.4 (Notes pages 78–87) '
• Introduction to generating functions
• Application: Fruit baskets
• Application: Dice
• Generating function manipulations

• Please watch this YouTube video that discusses slides 78-81.

• Prepare to answer the following questions in class. Feel free to use Wolfram Alpha or Mathematica to look at the coefficients of this generating function. Recall that the Mathematica command to find the coefficients of the generating function from class is:

  Series[1/(1-x)/(1-x^2)/(1-x^3),{x,0,98}]

  • Question 23. Exercise 3.3.2 on page 113.
  • Question 24. Suppose you roll three six-sided dice. Calculate the probability that the dice sum to fourteen. Now suppose you roll five six-sided dice. Calculate the probability the dice sum to fourteen. Which probability is higher?
  • Question 25. Exercise 3.3.3.
  • Question 26.
    

• Notes from Sections 3.5 and 3.4 (Notes pages 84–91) '
• Generating function manipulations
• Solving recurrence relations

• Project topic revision due today. Send the final version of your project statement by email before class.

• Solving recurrence relations
• Fibonacci generating function
• Products and powers of generating functions
• And their combinatorial interpretations
• Notes from external material (Notes pages 92–97) '
• Compositions of generating functions
• And their combinatorial interpretations

• Prepare for Assessment 3 on Standards 8 and 10.
• Prepare to answer the following questions in class.
  • Question 27. Complete Exercise 3.6.4. (In this question, give the explicit formula for a_n as a function of n using the recurrence relation techniques we learned in class.)
  • Question 28. Give a combinatorial counting question whose answer is the coefficient of x^k in the generating function (1+x)¹⁰/(1-2x)
  • Question 29.
    
• Spend some time thinking about your project.

• Assessment 3 on Standards 8 and 10.
• Homework Discussion
• Compositions of generating functions
• And their combinatorial interpretations
• Notes from Sections 2.4, 3.4, and 4.4 (Notes pages 98–105) '
• Integer partitions
• Integer partition bijections
• Standard Young Tableaux

• Spend some time thinking about your project. Bring what you have so far to class.
• Prepare to answer the following questions in class.
  • Question 30. Calculate the generating function that counts how many ways are there to take a line of n soldiers, break them into non-empty platoons, and choose some (possibly empty) subset of each platoon to be on "night watch". (Consider compositions of generating functions.) For an extra challenge, calculate this number as a function of n.
  • Question 31. Suppose you have an unlimited supply of black building blocks of height 1 and an unlimited supply of red, orange, yellow, green, blue, and purple building blocks of height 2. Calculate the generating function that counts the number of ways to build a tower of height n? (Consider compositions of generating functions.) For an extra challenge, calculate this number as a function of n.

• Homework Discussion
• Integer partition bijections
• Remainder of class: Poster Work Day. Prof. Chris will come around to talk to you about your project.

• Prepare for Assessment 4 on Standards 9 and 11.
• Prepare for any reassessments you would like to attempt.

• Assessment 4 on Standards 9 and 11.
• You can also take up to two reassessments during class time.
• Professor Chris is out of town this week. Professor Zeng will administer the assessment and the reassessments on Monday November 12.

• This week is devoted to working on your project. Spend some time working to attack your project statement. By Monday the 19th you should have counted at least some part of your project and formulated a plan for counting the rest.
• Professor Chris is out of town at a research event this week and we will not meet in person on Wednesday, November 14; use this time to work on your project

• Prepare to answer the following questions in class.
  • Question 32. (a) Determine the generating function for the number of partitions of n such that there are at most two parts of the same size. [For example, 511 is OK, but 4111 is not allowed since 1 appears thrice.] (b) Determine the generating function for the number of partitions of n such that the parts are all of size equal to a power of two. [For example: 84422 is OK, but 744221 is not because 7 is not a power of two.]
  • Question 33. Solve Exercise 4.4.2. You are given the partition z₁+z₂+…+z_k of n, and you now want to investigate the conjugate partition y₁+y₂+…. Try to determine a rule that tells you the value of y_i, the i-th part of the conjugate partition, as some function of the z-values. (Instead of appealing directly to the Ferrers diagram.)
    Make sure to explain clearly why your rule works.

• Homework Discussion
• Self-conjugate partitions
• Standard Young Tableaux
• Notes from Section 4.1 PLUS additional material (Notes pages 106–114) '
• Catalan numbers
• Catalan number interpretations
• (Handy handout about Catalan numbers)
• (link to Stanley's Catalan Interpretations)

• Prepare for Assessment 5 on Standards 12 and 13.
• Prepare to answer the following questions in class.
  • Question 34. 4.4.3
  • Question 35. Recall that a Dyck path of length n is a lattice path from (0,0) to (n,n) that stays above the line y=x.)
    1. Find and list the 14 Dyck paths of length 4 and the 14 multiplication schemes for 5 variables.
    2. Use the Catalan bijections from class to determine which Dyck path corresponds to which multiplication scheme.

• Assessment 5 on Standards 12 and 13.
• The "meta" method for Catalan numbers
• A formula for the Catalan numbers
• Notes on Combinatorial Statistics (Notes pages 115–125) '
• Combinatorial statistics
• Descent statistic des(π)
• Inversion statistic inv(π)
• Major statistic maj(π)

• Prepare to answer the following questions in class.
  • Question 36. Use a bijection to show that sequences 1≤ a₁≤ a₂≤ … ≤ a_n of length n, where each a_i≤ i are also counted by the Catalan number C_n. For example, when n=3, the five sequences are 111, 112, 113, 122, and 123. [Hint: Look at the boxes to the left of a Dyck path.]
  • Question 37. Another family of combinatorial objects that is counted by the Catalan numbers are rooted plane trees. (See Section 1.6 of the Handy handout about Catalan numbers.) Combinatorially show that these objects satisfy the generating function equation C(x)=1+xC(x)². (This will be a "meta" argument similar to the ones from class.)

• (no new notes)
• q-analogs
• q-factorial
• q-binomial coefficients
• PDF about q-analogs

• Prepare to answer the following questions in class.
  • Question 38.
    1. Calculate des(π), inv(π), and maj(π) for π=963852741.
    2. Let π be an n-permutation with reverse σ. How is inv(π) related to inv(σ)?
  • Question 39. Two combinatorial interpretations of the q-binomial coefficients are given on page 124 of the course notes.
    1. Show that for permutations π of the multiset {1²,2³},
       
    2. Show that for the set of lattice paths P from (0,0) to (2,3)$,
       
  • Question 40.
    

• Catch-up with topics
• Depending on the time available: Notes (Notes pages 126–143) '
• Research topics in algebraic combinatorics

• Prepare for Assessment 6 on Standards 14–17.

• Assessment 6 on Standards 14–17.
• Remainder of class: Reassessments or Poster Work Day

•••••• Please fill out the college-wide course evaluations, distinct from the course evaluations that will be given out in class. Thank you for your feedback! ••••••

• Continue work on Poster. Bring what you have to class so far.

• In-class project work day

• Complete work on your poster.
• Bring it to class at 4:45pm.

• Poster Session

• Podcast recording session

• Question and Answer Day

• Exam 2

•••••• Please fill out the college-wide course evaluations, distinct from the course evaluations that will be given out in class. Thank you for your feedback! ••••••