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ma01c09rec4-2 - Ma1c Analytic Recitation 1 Info I am Alden...

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Ma1c Analytic Recitation 4/2/09 1 Info I am Alden Walker. My office hour is Fridays 1–2pm. The best way to contact me is by email at [email protected] I will hopefully post notes such as these every week on my website for this course, which you can find under “teaching” on my website http://www.its.caltech.edu/~awalker . Homework is due Mondays at 10am in the boxes outside the math office. You get one free extension of one week. It is nice if you tell me beforehand so that we don’t go looking for your “missing” homework. The grading in the course is 40% final, 30% midterm, and 30% homework, so it’s heavy on the final. You cannot skip the final. 2 Interesting Problem Consider two numbers. Let T n be the number of full rooted binary trees with n + 1 leaves. Let W n be the number of walks in the integer lattice in R 2 from (0 , 0) to ( n, n ) which do not go below the line y = x . Why is T n = W n ? Here is how you get a walk from a tree. I think you will see how to go backwards, although it is a little more tricky. Label the edges of a tree in the following way: on every edge going down and to the left, put a “U”, and on every edge going down and to the right, put a “R“. Now, do a depth-first traverse of the tree, writing down the labels of the edges as you pass them. You do not write down any edges after the first time you traverse them (or, say, only write down a label if you are going down the edge), and your depth-first traverse always goes to the left first. Notice that at the end of this you have a string like “UURURR.” Translate this into a walk in the integer lattice by going up for every “U” and to the right for every “R.” You go left first, so you always have more U’s than R’s, which means your walk does not go below the line y = x . To go backward, it is possible to reconstruct the tree from the traverse labels, but it requires a little thought. The numbers T n = W n are called the Catalan numbers, and they count many interesting things. There is a nice formula which says that T n = W n = 1 n +1 ( 2 n n ) . 3 Open Sets A bunch of questions on your homework have to do with open sets in R n . I think you’ll find these questions easier than the continuity questions, at least intuitively, but it is important, mostly for problem 2, to be careful when proving things are open. Here are some definitions—there are tons of various words thrown
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