MIT6_042JS10_lec31

MIT6_042JS10_lec31 - Mathematics for Computer Science...

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1 Albert R Meyer, April 23, 2010 lec 11F.1 Mathematics for Computer Science MIT 6.042J/18.062J Generating Functions Albert R Meyer, April 23, 2010 Infinite Geometric Sum S(x) ::= 1+ x + x 2 + + x n + lec 11F.2 xS(x) = x + x 2 + + x n + Albert R Meyer, April 23, 2010 Infinite Geometric Sum S(x) ::= 1+ x + x 2 + + x n + S(x) xS(x) = lec 11F.3 1 xS(x) = x + x 2 + + x n + Albert R Meyer, April 23, 2010 Ordinary Generating Functions The ordinary generating function for the infinite sequence ! g 0 , g 1 , g 2 , , g n , " is the power series: G(x) = g 0 +g 1 x+g 2 x 2 + +g n x n + lec 11F.5 g n g n Albert R Meyer, April 23, 2010 lec 11F.6 “corresponds to” Infinite Geometric Sum = 1 1 x Albert R Meyer, April 23, 2010 Bags of fruit lec 11F.8 In how many ways can we fill a bag with n fruits given the following constraints? At most 2 oranges . Any number of apples . Any number of bananas that only come in bunches of 3.
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2 Albert R Meyer, April 23, 2010 Bags with n = 4 fruits • 0 oranges , 1 apple , 3 bananas • 0 oranges , 4 apples , 0 bananas • 1 orange , 0 apples , 3 bananas • 1 orange , 3 apples , 0 bananas • 2 oranges , 2 apples , 0 bananas lec 11F.9 Number of 4-fruit bags: 5 Albert R Meyer, April 23, 2010 At most 2 oranges lec 11F.10 # ways to pick k oranges Albert R Meyer, April 23, 2010 There is only 1 way to pick a bag of k apples: a k = 1 Any number of apples lec 11F.11 Albert R Meyer, April 23, 2010 Substituting x k for x lec 11F.12
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