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A2 - Math 239 Assignment 2 Due Friday May 21 1 Express the...

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Due: Friday, May 21 Math 239 Assignment 2 1. Express the following coefficients as summations involving only ordinary binomial coef- ficients. (a) (4 points) [ x n ](1 - 2 x ) - m (1 - x 2 ) m . (b) (4 points) [ x n ](1 + 2 x - 3 x 2 ) - 4 . (c) (4 points) [ x n ](1 + 2 x ) - 5 (1 - x 3 ) m . 2. Let x 1 , . . . , x k be variables. A monomial in these variables is a product x m 1 1 · · · x m k k where m i ’s are non-negative integers. (For example monomials in x 1 and x 2 include 1, x 5 2 and x 2 1 x 2 .) The degree of a monomial is the sum of the exponents—in this case m 1 + · · · + m k . Let N denote the set of non-negative integers. (a) (2 points) Define a weight function on N k such that the number of elements in N k with weight m is equal to the number of monomials in x 1 , . . . , x k with degree exactly m . (b) (3 points) Compute the generating series for N k . (c) (3 points) Use the binomial theorem to compute the number of monomials in k variable with degree exactly n . 3. Let S be the set of all subsets of { 0 , . . . , n - 1 } and define the weight of a subset to be the sum of its elements. Let T be the subset of S consisting of the subsets that do not contain n - 1. Let Φ
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