mid_sample - 111 Determine the generating function for A As...

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MATH 239 SAMPLE Mid-Term Exam — Winter 2007 1. (a) [4 marks] Determine [ x 7 ]( x 3 (1 - 2 x ) 2 - 2(1 + x ) 9 ). (b) [4 marks] Prove that 30 X i =10 ± i 10 ² = ± 31 11 ² . (Hint: consider (1 - x ) - 1 (1 - x ) - k = (1 - x ) - ( k +1) .) 2. [8 marks] Find the generating function for the number of compositions of a non-negative integer n with an odd number of parts, where no part is divisible by three. (Note the number of parts is not fixed.) 3. Let A ( x ) be a formal power series, where A ( x ) = X n 0 a n x n and A ( x ) = x (2 + 2 x - 3 x 2 ) 1 - 2 x - x 2 + x 3 . (a) [7 marks] Determine a linear recurrence with initial conditions for the sequence a n of coefficients of A ( x ). (b) [5 marks] Find a 0 , . . . , a 4 . 4. (a) [5 marks] Let S be the set of all binary strings in which every block of zeros has even length, and every block of ones has length at most three. Write a decomposition that uniquely creates the elements of S . (b) [7 marks] Let A be the set of binary strings defined by A = ( { ± } ∪ { 0 }{ 000 } * )( { 11 }{ 111 } * { 0 }{ 000 } * ) * ( { ± } ∪ { 11
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Unformatted text preview: }{ 111 } * ) } . Determine the generating function for A . As usual, the weight of a string is its length. Express your answer as p ( x ) q ( x ) , where p ( x ) and q ( x ) are polynomials or products of poly-nomials. 5. [8 marks] Let the sequence b n be defined by b = 1, b 1 = 1, and b n +2 = 4 b n +1-4 b n for all n ≥ 2 . Solve this recurrence relation to obtain a closed form expression for b n . 6. [6 marks] Draw all the non-isomorphic trees on 7 vertices and briefly describe why no two are isomorphic. 7. [6 marks] Let G be a connected graph. Suppose P and Q are two paths in G having no vertices in common. Let the length ( = number of edges) of P be p and the length of Q be q . Prove that G has a path whose length is at least 1 + ( p/ 2) + ( q/ 2)....
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