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Homework assignments
(MATH 3166-001, Spring 2013)
Instructor: Gábor Hetyei Last update: Wednesday, April 17, 2013

Disclaimer: The information below comes with no warranty. If, due to typographical error, there is a discrepancy between the exercises announced in class and the ones below, or this page is not completely up to date, the required homework consists of those exercises which were announced in class. Check for the time of last update above. If, by my mistake, a wrong exercise shows up below, I will allow you extra time to hand in the exercise that was announced in class. If, however, exercises are missing because this page is not up to date, it is your responsibility to contact me before the due date. (No extra time will be allowed in that case.) This page is up to date if the last update happened after the last class before the next due date.

The deadlines do not apply to the Bonus questions, which expire only once we solve them in class, or on April 22 at latest.

Notation: 1.4/16a means exercise 16, part a, in chapter 1, section 4.

No. Date due: Problems:
12 We Apr 24 8.1/1,4.
Board problem: Complete the proof of the fact that the join operation in a lattice is associative.
Bonus problem: 4.4/9a. (This was assigned as a regular question last week, I changed it to a bonus question.) This is the only bonus problem that is due on Wednesday April 24, all other bonus questions are due on Monday April 22.
11 We Apr 17 4.3/6,9,12 (final answer is in the back, you have to show your workl!);   4.4/2, 9a.
Board problem: Using the same method as for 4.3/12, find a closed formula for f(n)=13+23+...+n3
10 We Apr 10 4.2/1a, 6b   4.3/1, 4cd (use ab), 8, 9, 12. I also distributed a bonus question due April 10. The score for this bonus question will be added tou your Test 2 score. Note added on Monday, April 8, 2013: Today we postponed the deadline of 4.3/9 and 4.3/12 till April 17.
9 We Mar 20 4.1/2a, 5, 4, 6, 8 (finish argument given in the back of the book), 13.
Bonus questions:
  1. Write a generating function equation for the Catalan numbers, solve it and deduce the explicit formula for the Catalan numbers. (Similar to the argument given in the book for the triangulations, but you have to shift all indices by two.)
  2. Prove that the Catalan number Cn is the number of circular arrangements of n+1 copies of 1 and n copies of -1. (Define Cn as the number of lattice paths from (0,0) to (2n,0), using steps of the form (1,1) and (1,-1), which never go below the x axis.)
8 We Mar 20 3.6/2bd;   3.5/1be, 3ab.
7 We Mar 13 3.3/3be, 4, 8, 9b;   3.4/2,5,10.
6 We Feb 27 3.1/2,4;   3.2/1bc, 3, 4a, 9a, 10, 11.
Bonus question: 3.2/9b.
5 We Feb 13 2.3/4,6,9,11;   2.4/8,10 (you may use Ferrer's diagrams instead of type vectors), 12 (hint: use the answer to 11 from the back of the book).
4 We Feb 6 2.1/3,4d;   2.2/4ac,8,9.
Bonus questions:
  1. 2.2/5
  2. Extend Vandermonde's formula (formula (2.5) in section 2.1) to the case when m and n are not integers.
3 We Jan 30 1.4/2,6,10,16.
2 We Jan 23 1.2/4,6,8,10;   1.3/4,12.
1 We Jan 16 1.1/6,8,10,16.
Bonus question: 1.1/12. Note: The book does not make it clear, but my understanding is that sets in a Venn diagram are represented by circles, not arbitrary (even concave) shapes. This reading of the question makes it hard, and that's why I consider it bonus.