Math 239 - Assignment 2

Math 239 - Assignment 2 - 1 with respect to w . Express...

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MATH 239 Assignment 2 This assignment is due on Friday, Jan 21 2011, at noon in the drop boxes outside MC 4067. 1. Let A ( x ) = i 0 a i x i and B ( x ) = i 0 b i x i be formal power series with rational coefficients, and suppose that both have inverses. Let A - 1 ( x ) and B - 1 ( x ) denote the inverses of A ( x ) and B ( x ) respectively. (a) Give examples to show that A ( x ) + B ( x ) does not necessarily have an inverse. (b) Prove that A ( x ) + B ( x ) has an inverse if and only if A - 1 ( x ) + B - 1 ( x ) has an inverse. (c) Prove that there do not exist any A ( x ) and B ( x ) such that the inverse of A ( x )+ B ( x ) is equal to A - 1 ( x ) + B - 1 ( x ). 2. (a) Determine the coefficient [ x n ] x 2 k (1 - x 20 ) k (1 - x 2 ) - k . (b) Determine the coefficient [ x n ](1 + 2 x ) - 5 (1 - x 3 ) - m . 3. Consider the set N 1 = { 1 , 2 , 3 ,... } of positive integers, with the natural weight function w ( σ ) = σ . The generating function for N 1 with respect to w is x 1 - x . Find the generating function for each of the following subsets of N
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Unformatted text preview: 1 with respect to w . Express each generating function (except in (d)) as a rational function , that is, in the form p ( x ) q ( x ) where p ( x ) and q ( x ) are polynomials in x . (In part (d) you may leave your answer as a sum of rational functions.) (a) The set of positive integers that are divisible by 6. (b) The set of positive integers that are congruent to 1 (mod 20). (c) The set of positive integers that are divisible by 4. (d) The set of positive integers that are not divisible by 6 and are not congruent to 1 (mod 20). (e) The set of positive integers that are not divisible by 6 and not divisible by 4. 4. Find the number of compositions of n with k parts, where k 1, in which each part is at least 2 and at most 8....
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