Announcements

• Final exam will be handed out on November 24, and will be due on Monday, December 8. You may freely refer to Aigner's and Stanley's texts as well as your notes and homework. You may also, as usual, ask me for hints. You may not use any other sources, and are not allowed to discuss the contents of this exam with anyone else until after you turn it in.
• Correct second proof of the result about sub-multisets.

Homework

• Homework 1, due September 12.
  • Read sections 1.1-1.4 of Aigner and (if you have access to them) sections 1.3-1.5 of Stanley.
  • Aigner Chapter 1, problems 5 (use the hint if necessary), 6 (you may use the results from later sections; provide both an "inductive" proof and a "direct" proof), 12 (easy), 29, 31 (hard. First, identify a partition with an n-sub-multiset of [k] a sequence of length n, with the terms in the positions of the i'th block equal to i. Additional hints provided upon request), 34.
• Homework 2, due September 26.
  • Read sections 1.4-1.6 of Aigner and (if you have access to them) sections 1.3-1.5 of Stanley.
  • Aigner Chapter 1, problems 38 (use induction), 41, 45, 48 (use the hint), 49, 52 (see page 1).
• Homework 3, due September 29.
  • Read sections 1.5-1.6, 2.1-2.4 of Aigner.
  • Aigner Chapter 1, problems 53, 54, 58, 63, 72 (use lattice paths), 74 (how many ways are there to choose an ordered set of k linearly independent vectors in V? How many ways are there to choose an ordered basis in a k-dimensional subspace of V?).
• Homework 4, due October 3.
  • Read sections 2.1-2.5, 3.1 of Aigner.
  • Aigner Chapter 2, problems 4, 5, 19, 20, 22, 30 (misprint: 2k should be 2^k), 39.
• Homework 5, due October 10.
  • Read sections 3.1-3.2 of Aigner.
  • Aigner Chapter 3, problems 5, 6, 9 (consider the first time when the number of A's and B's is equal), 13 (use the hint, or convolution), 19 (use series from 2.1), 26.
• Homework 6, due October 17.
  • Read sections 3.2-3.4 of Aigner.
  • Aigner Chapter 3, problems 29, 30(a,b), 33 (combinatorial interpretation optional), 34, 44, 46.
• Homework 7, due October 24.
  • Read sections 3.3, 3.4, 5.1 of Aigner.
  • Aigner Chapter 3, problems 58, 59, 66, 69, 78, 79 (one method is to use a familiar bijection).
• Homework 8, due October 31.
  • Read sections 5.1, 5.2 of Aigner and (if you have access to them) sections 3.1, 3.2, 3.6-3.8 of Stanley.
  • Aigner Chapter 5, problems 2, 4, 11, 12, 16 (you may assume that n is even).
• Homework 9, due November 7.
  • Read section 5.2, of Aigner and (if you have access to them) sections 3.1, 3.2, 3.6-3.10 of Stanley.
  • Aigner Chapter 5, problems 26, 27, 28, 29, 30, 35.
• Homework 10, due November 14.
  • Read sections 5.2, 5.3 of Aigner and (if you have access to them) sections 3.6-3.10 of Stanley.
  • Aigner Chapter 5, problems 31, 32, 37 (re-prove the result from class using induction and the q-binomial theorem), 38, 43, 45.
• Homework 11, due November 21.
  • Read sections 5.3, 5.4 of Aigner.
  • Aigner Chapter 5, problems 50, 51, 52(a,b), 53, 58.
• Homework 12, due December 1.
  • Read sections 5.4, 8.1-8.2 of Aigner.
  • Aigner Chapter 5, problems 64, 66, 69, 70, 76, and the exercise assigned in class.
• Optional Homework 13, due December 8. If you turn it in, its score will be added to the total homework score.
  • Read sections 8.1-8.5 of Aigner.
  • Aigner Chapter 8, problems 5, 6, 10, 12, 15, 17.