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# ass5 - x n(1-2 x-2 and using this compute the average 4 Let...

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Due: Friday, June 8 Math 239 Assignment 5 1. (a) Find the generating function for the set S of binary strings where each block of 1’s has odd length and no blocks of zeros has length greater then three. (b) Derive a recurrence for the coefficients of the generating function, and use this to com- pute the first five coefficients. 2. Let L denote the set of binary strings that do not contain 0110 as a substring. Using the recurrence method, compute the generating function for L . 3. We have the block decomposition of binary strings {0,1} * = 1 * (0 + 1 + ) * 0 * . (Here A + : = AA * .) Let a mnk denote the number of binary strings containing m 1’s, n 0’s and k blocks of 1’s. Let G ( x , y , u ) be the corresponding generating function, where 0 7→ x and 1 7→ y and u counts blocks of 1’s. (a) (2 points) Show that G ( x , y , u ) = 1 + uy 1 - y 1 1 - x 1 - x uy 1 - y 1 1 - x = 1 - y + uy (1 - x )(1 - y ) - uxy (b) (5 points) Show that the average number of blocks of 1’s in a binary string of length n is 2 - n [ x n ] x - x 2 (1
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Unformatted text preview: x n ](1-2 x )-2 and, using this, compute the average. 4. Let A ( x ) and B ( x ) be series with constant terms equal to 1 such that B ( x ) = A ( x ) 3 . Note that if C ( x ) is the series ∑ n ≥ c n x n , then its derivative C ( X ) is ∑ n ≥ ( n + 1) c n + 1 x n . (a) (2 points) Using the usual rules from Calculus, show that A ( x ) B ( x ) = 3 A ( x ) B ( x ). (b) (2 points) Assuming that [ x k ] A ( x ) = a k and [ x k ] B ( x ) = b k , compute formulas for the coefﬁcient of x n in A ( x ) B ( x ) and in A ( x ) B ( x ). (c) (5 points) Apply the previous material with B ( x ) = (1-x )-1 to compute the ﬁrst three coefﬁcients of (1-x )-1/3 . 5. (5 points) Solve the recurrence equation: b n-3 b n-1-4 b n-2 + 12 b n-3 = where b = 1, b 1 = 7, b 2 = 9....
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