# A4b - C&amp;amp;amp;O 330 - ASSIGNMENT #4 DUE FRIDAY, 5...

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Unformatted text preview: C&amp;O 330 - ASSIGNMENT #4 DUE FRIDAY, 5 NOVEMBER AT 10:31AM (1) (15 points) A rise in a sequence 1 p is a pair ( j , j +1 ) such that j &lt; j +1 . Prove that the number of permutations on n symbols with exactly k rises is bracketleftbigg u k x n n ! bracketrightbigg u- 1 u- e ( u- 1) x . [ Hint: Use the Maximal Decomposition Theorem and the Permutation Lemma.] (2) (15 points) Prove that the number of permutations of length n, with precisely k maximal &lt;-substrings having length greater than or equal to 2 , is bracketleftbigg u k x n n ! bracketrightbigg ( cosh( zx )- z- 1 sinh ( zx ) )- 1 where z = 1- u. [ Comment: Recall that cosh( y ) = k y 2 k (2 k )! and sinh ( y ) = k y 2 k +1 (2 k +1)! . ] (3) (15 points) Let D ( x 1 , x 2 , . . . ) be the ordinary generating series for the number d ( k 1 , . . . , k n ) of sequences with k i occurrences of i for i = 1 , . . ., n such that adjacent symbols in the sequence are not equal....
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## This note was uploaded on 11/18/2010 for the course CO 330 taught by Professor R.metzger during the Spring '05 term at Waterloo.

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A4b - C&amp;amp;amp;O 330 - ASSIGNMENT #4 DUE FRIDAY, 5...

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