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Homework?
Maple?
Final exam vs. final project
Return exam
Give overall comments:
1. Read everything before you start. Some of you didn't realize that
when I ask for a formula for a sequence, I want a closed form
formula, not a recurrence relation!
But I explained this at on the
last page of the exam.
2. Show work
3. Check work
4. Using Maple
5. Don't write 1/(x1) just because Maple does!
Final answers
should be in simplest, most helpful form
Describe breakdown of grades: 100, 97, 96, 91, 85, 68, 67, 66
TODAY:
Catalan numbers and parenthesizations of products
Catalan numbers and triangulations of polygons
The “short” recurrence for Catalan numbers
Parenthesizations of products
Let P_n be the number of ways to parenthesize a product of n
things.
E.g., P_1 = P_2 = 1, but P_3 = 2 (since abc can be parenthesized as
either (ab)c or a(bc).
Claim: For n
≥
3, P_n = P_2 P_{n2} + P_3 P_{n3} + …
+ P{n2} P_2.
This should remind you of … the recurrence relation for Catalan
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This note was uploaded on 02/13/2012 for the course MATH 491 taught by Professor Propp during the Fall '08 term at UMass Lowell.
 Fall '08
 Propp
 Algebra, Combinatorics

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