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hw4 - n-gon as the base and let u n be the number of...

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Math 146: Algebraic Combinatorics Homework 4 This is a half problem set due Friday, April 23. Do problems 2.7, 2.26 and 2.27(all but part a), in addition to the following problems: GK4.1. Another occurrence of Catalan numbers. Let t n be the number of ways to divide a convex n -gon into n - 2 triangles. For example, t 5 = 5, as follows: (a) Prove the recurrence t n = n - 1 k = 2 t k t n - k + 1 for n 3. (b) Use part (a) to solve for the generating function. (c) Use part (b) to solve for t n itself. (d) A counting argument: Mark one side of the
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Unformatted text preview: n-gon as the base, and let u n be the number of triangulations with in which a side other than the base has been marked. Let v n be the number of triangulations in which one of the edges of the triangulation (either a diagonal or a side, including the base) has been turned into an arrow. Show that u n = ( n-1 ) t n v n = ( 4 n-6 ) t n . Then show that u n + 1 = v n . Use these three equations to reprove part (c) directly. 1...
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