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!

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 and December 1. 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 2abde, 2c, 4abc, 4d, 12, 16, 20, and 25 in section 6.1 and problems 1, 2, 11abcd, 11e, 30, 32, and 43 in section 6.2, but submit only your solutions of the underlined problems.

In addition, solve the following problem. There is no need to submit your solution.

Part 1. For $n\geq 0$, let $C_{n}$ be the $n$'th Catalan number, defined as the number of words made 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 $n\geq 1$, $C_{n}=C_{0}C_{n-1}+C_{1}C_{n-2}+C_{2}C_{n-3}+\ldots +C_{n-1}C_{0}$.

Hint. If $w$ is such a word, let $k\geq 1$ be the smallest such that if you cut $w$ after $2k$ letters, then the number of $a$'s before the cut is equal to the number of $b$'s before the cut. What can you say about the part of $w$ before the cut, and the part of $w$ after the cut?

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 $n\geq 2$, $D_{n}=D_{1}D_{n-1}+D_{2}D_{n-2}+D_{3}D_{n-3}+\ldots +D_{n-1}D_{1}$.

Hint. In the last matrix multiplication to be carried out, you'd be multiplying the product of $k$ of the matrices with the product of $n-k$ of the matrices.

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 $n\geq 3$, $E_{n}=E_{2}E_{n-1}+E_{3}E_{n-2}+E_{4}E_{n-3}+\ldots +E_{n-1}E_{2}$.

Hint. Choose one edge of the $n$-gon, and consider the triangle on top of it, what's to its left, and what's to its right.

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

Homework Assignment 9 Solution

2) Build a generating function for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_{r}} , the number of integer solutions to the following equations.

c) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e_{1}+e_{2}+e_{3}+e_{4} = r, 2\leq e_{i} \leq 7, e_{1}} even Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e_{2} } odd.

A) For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e_{3}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e_{4}} together, we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_{1}(x) = (x^2+x^3+x^4+x^5+x^6+x^7)^2 }

For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e_{1}} , we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_{2}(x) = (x^2+x^4+x^6) }

For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e_{2}} , we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_{3}(x) = (x^3+x^5+x^7) }

In total, we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x) = (x^2+x^4+x^6)(x^3+x^5+x^7)(x^2+x^3+x^4+x^5+x^6+x^7)^2 }

4 d) Three different boxes with at most five objects in the first box

A) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e_{1}+e_{2}+e_{3} = r} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0\leq e_{1} \leq 5}

For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e_{1}} , we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1+x+x^2+x^3+x^4+x^5) }

Note that for the other two, we don't have a limit, so:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{k=0}^{\infty}x^{k} = \frac{1}{1-x} }

so, in total we have: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x) = (1+x+x^2+x^3+x^4+x^5)(\frac{1}{1-x})^2 }

16) Find a generating function for the number of integers between 0 and 999,999 whose sum of digits is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r } .

A) For each integer in the range, we can express them as a six digit number, including 0's in the front (5 will be 000005). So this amounts to finding:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e_{1}+e_{2}+e_{3}+e_{4}+e_{5}+e_{6} = r , 0 \leq e_{i} \leq 9 , 1 \leq i \leq 6 }

So, we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x) = (1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9)^6 }

Section 6.1 2) Find the coefficient of $x^{r}$ in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x^5 + x^6 + x^7 +...)^8}

A)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x^5 + x^6 + x^7 +...)^8 = (x^5(1+ x + x^2 ...))^8 = x^{40} \frac{1}{(1-x)^8} }

The coefficient of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^r } in this will be of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^{r-40} } in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1-x)^{-8} } , which is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \binom{(r-40) + 8 - 1}{r-40} }