15-344/Homework Assignment 9

DBN: Classes: 2015-16: 15-344 / Navigation Welcome to Math 344! Edits to the Math 344 web sites no longer count for the purpose of good deed points. # Week of... Notes and Links 1 Sep 14 About This Class, Day One Handout, Tuesday, Hour 3 Handout, Thursday, Tutorial 1 Handout 2 Sep 21 Tuesday, Tutorial 2 Page 1, Tutorial 2 Page 2, Thursday, HW1 3 Sep 28 Tuesday, Class Photo, Tutorial 3 Page 1, Tutorial 3 Page 2, Thursday, HW2 4 Oct 5 Tuesday, Drawing $n$-cubes, Thursday, HW3 5 Oct 12 Tuesday, Tutorial Handout, Thursday, HW4 6 Oct 19 Tuesday, Tutorial Handout, Thursday 7 Oct 26 Term Test on Tuesday, Dijkstra Handout,Thursday,HW5 8 Nov 2 Tuesday, Tutorial Handout, Thursday, HW6, Sunday November 8 is the last day to drop this class 9 Nov 9 Monday-Tuesday is UofT Fall Break, Thursday, HW7 10 Nov 16 Tuesday, Thursday, HW8 11 Nov 23 Tuesday, Thursday, HW9 12 Nov 30 Tuesday, Thursday, HW10 13 Dec 7 Tuesday, FibonacciFormula.pdf, semester ends on Wednesday - no class Thursday F Dec 11-22 The Final Exam Register of Good Deeds Add your name / see who's in!

In Preparation

The information below is preliminary and cannot be trusted! (v)

This assignment is due at the tutorials on Thursday December 3. Here and everywhere, neatness counts!! You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.

Reread sections 6.1-6.2 of our textbook, and your notes for November 26. Remember that reading math isn't like reading a novel! If you read a novel and miss a few details most likely you'll still understand the novel. But if you miss a few details in a math text, often you'll miss everything that follows. So reading math takes reading and rereading and rerereading and a lot of thought about what you've read. Also, preread chapter 7, just to get a feel for the future.

Solve problems ... in section ..., but submit only your solutions of the underlined problems.

In addition, solve the following problem, but submit only your solutions to parts 2, 3, and 4:

Part 1. For $n\geq 0$, let $C_{n}$ be the $n$'th Catalan number, defined as the number of sequences of $n$ $a$'s and $n$ $b$'s in which in every initial segment there are at least as many $a$'s as $b$'s. Prove that for $n\geq 1$, $C_{n}=C_{0}C_{n-1}+C_{1}C_{n-2}+C_{2}C_{n-3}+\ldots +C_{n-1}C_{0}$.

Part 2. For $n\geq 1$, let $D_{n}$ be the number of different ways of computing the product $A_{1}A_{2}\cdots A_{n}$ of $n$ matrices. Prove that for $n\geq 2$, $D_{n}=D_{1}D_{n-1}+D_{2}D_{n-2}+D_{3}D_{n-3}+\ldots +D_{n-1}D_{1}$.

Part 3. Let $E_{2}=1$ and for $n\geq 3$ let $E_{n}$ be the number of triangulations of a convex $n$-gon using non-crossing diagonals. Prove that for $n\geq 3$, $E_{n}=E_{2}E_{n-1}+E_{3}E_{n-2}+E_{4}E_{n-3}+\ldots +E_{n-1}E_{2}$.

Part 4. Use the above three assertions to prove that for any $n\geq 0$, $C_{n}=D_{n+1}=E_{n+2}$.

Dror's notes above / Students' notes below