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184a_01w_2e

# 184a_01w_2e - Math 184A Second Exam(125 points 26 February...

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Math 184A Second Exam (125 points) 26 February 2001 Please put your name and ID number on your exam. If you are not using a blue book, put your name on every page. The exam is CLOSED BOOK, but ONE SHEET of notes is allowed. You must show your work to receive credit. 1. (45 pts.) (a) Compute the rank of the permutation 1 , 5 , 3 , 4 , 2 when the permutations of { 1 , 2 , 3 , 4 , 5 } are listed in direct insertion order. (b) What permutation immediately follows 1 , 5 , 3 , 4 , 2 in direct insertion order? (c) What permutation immediately follows 1 , 5 , 3 , 4 , 2 in lex order? 2. (20 pts.) Find the unlabeled rooted plane tree with 7 leaves and rank 60. b 1 = b 2 = 1 , b 3 = 2 , b 4 = 5 , b 5 = 14 , b 6 = 42 , b 7 = 132 . 3. (30 pts.) Let C n ( k ) be the number of times position k is changed in the Gray code for n -long vectors of zeroes and ones given in the text. You may use the following fact without proving it: C n ( k ) = ( 1 if k = 1, 2 C n - 1 ( k - 1) if 1 < k n , 0 if k > n . (a) Tabulate values of C n ( k ) for 1 k n 4. (b) State and prove a simple formula for C n ( k ) that does not involve a recursion.
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