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184a_03f_2e

# 184a_03f_2e - a n = 2 n n-long sequences of zeroes and ones...

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Math 184A Second Exam 24 November 2003 Please put your name and ID number on your blue book. The exam is CLOSED BOOK, but you may have a page of notes. Calculators are NOT allowed. You must show your work to receive credit. 1. (16 pts.) An oriented graph is a simple graph in which each edge has been given a direction. In other words, given vertices x and y with x negationslash = y , exactly one of the following is true: There is no edge between x and y . There is an edge from x to y . There is an edge from y to x . Obtain formulas for (a) the number of oriented graphs with vertex set n ; that is, the number of n -vertex oriented graphs; (b) the number of n -vertex oriented graphs having exactly k edges. 2. (a) (2 pts.) Sketch the simple graph G = ( V, E ) where V = { a, b, c, d, e } E = braceleftbig { a, b } , { a, c } , { a, d } , { b, c } , { d, e } bracerightbig . (b) (6 pts.) Compute the chromatic polynomial P G ( x ). (c) (2 pts.) How many ways can G be properly colored if 5 colors are available? 3. (24 pts.) There are
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Unformatted text preview: a n = 2 n n-long sequences of zeroes and ones, including the empty sequence, and so A ( x ) = ∑ a n x n = (1 − 2 x )-1 . (You do not need to derive this.) Let f n be the number of such sequences that do not contain the pattern 11100. Let F ( x ) = ∑ f n x n . (a) Derive either of the two formulas A ( x ) = F ( x ) + A ( x ) x 5 F ( x ) A ( x ) = ∞ s t =0 ( F ( x ) x 5 ) t F ( x ) . (Both formulas are correct. Which you derive will depend on how you think about the problem.) (b) Using either of the formulas in (a) and the formula for A ( x ), Fnd polynomials P ( x ) and Q ( x ) so that F ( x ) = P ( x ) Q ( x ) ; for example, F ( x ) might be 7 23-x 9 . (c) Using (b) or otherwise, obtain a simple recursion for f n for n ≥ 5. Don’t worry about initial conditions. Final Exam in Center 113 END O± EXAM...
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