15-344/Homework Assignment 9: Difference between revisions
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'''Solve''' problems ... in section ..., but submit only your solutions of the underlined problems. |
'''Solve''' problems ... in section ..., but submit only your solutions of the underlined problems. |
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'''In addition,''' solve the following problem |
'''In addition,''' solve the following problem. There is no need to submit your solution. |
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'''Part 1.''' For <math>n\geq 0</math>, let <math>C_n</math> be the <math>n</math>'th Catalan number, defined as the number of |
'''Part 1.''' For <math>n\geq 0</math>, let <math>C_n</math> be the <math>n</math>'th Catalan number, defined as the number of words made of <math>n</math> <math>a</math>'s and <math>n</math> <math>b</math>'s in which in every initial segment there are at least as many <math>a</math>'s as <math>b</math>'s. Prove that for <math>n\geq 1</math>, <math>C_n=C_0C_{n-1}+C_1C_{n-2}+C_2C_{n-3}+\ldots+C_{n-1}C_0</math>. |
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''Hint.'' If <math>w</math> is such a word, let <math>k\geq 1</math> be the smallest such that if you cut <math>w</math> after <math>2k</math> letters, then the number of <math>a</math>'s before the cut is equal to the number of <math>b</math>'s before the cut. What can you say about the part of <math>w</math> before the cut, and the part of <math>w</math> after the cut? |
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'''Part 2.''' For <math>n\geq 1</math>, let <math>D_n</math> be the number of different ways of computing the product <math>A_1A_2\cdots A_n</math> of <math>n</math> matrices. Prove that for <math>n\geq 2</math>, <math>D_n=D_1D_{n-1}+D_2D_{n-2}+D_3D_{n-3}+\ldots+D_{n-1}D_1</math>. |
'''Part 2.''' For <math>n\geq 1</math>, let <math>D_n</math> be the number of different ways of computing the product <math>A_1A_2\cdots A_n</math> of <math>n</math> matrices. Prove that for <math>n\geq 2</math>, <math>D_n=D_1D_{n-1}+D_2D_{n-2}+D_3D_{n-3}+\ldots+D_{n-1}D_1</math>. |
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''Hint.'' In the last matrix multiplication to be carried out, you'd be multiplying the product of <math>k</math> of the matrices with the product of <math>n-k</math> of the matrices. |
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'''Part 3.''' Let <math>E_2=1</math> and for <math>n\geq 3</math> let <math>E_n</math> be the number of triangulations of a convex <math>n</math>-gon using non-crossing diagonals. Prove that for <math>n\geq 3</math>, <math>E_n=E_2E_{n-1}+E_3E_{n-2}+E_4E_{n-3}+\ldots+E_{n-1}E_2</math>. |
'''Part 3.''' Let <math>E_2=1</math> and for <math>n\geq 3</math> let <math>E_n</math> be the number of triangulations of a convex <math>n</math>-gon using non-crossing diagonals. Prove that for <math>n\geq 3</math>, <math>E_n=E_2E_{n-1}+E_3E_{n-2}+E_4E_{n-3}+\ldots+E_{n-1}E_2</math>. |
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''Hint.'' Choose one edge of the <math>n</math>-gon, and consider the triangle on top of it, what's to its left, and what's to its right. |
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'''Part 4.''' Use the above three assertions to prove that for any <math>n\geq 0</math>, <math>C_n=D_{n+1}=E_{n+2}</math>. |
'''Part 4.''' Use the above three assertions to prove that for any <math>n\geq 0</math>, <math>C_n=D_{n+1}=E_{n+2}</math>. |
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Revision as of 16:01, 26 November 2015
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. There is no need to submit your solution.
Part 1. For , let be the 'th Catalan number, defined as the number of words made of 's and 's in which in every initial segment there are at least as many 's as 's. Prove that for , .
Hint. If is such a word, let be the smallest such that if you cut after letters, then the number of 's before the cut is equal to the number of 's before the cut. What can you say about the part of before the cut, and the part of after the cut?
Part 2. For , let be the number of different ways of computing the product of matrices. Prove that for , .
Hint. In the last matrix multiplication to be carried out, you'd be multiplying the product of of the matrices with the product of of the matrices.
Part 3. Let and for let be the number of triangulations of a convex -gon using non-crossing diagonals. Prove that for , .
Hint. Choose one edge of the -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 , .
| Dror's notes above / Students' notes below |
