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# lec07 - AMS 301 Lecture 7 AMS Ning SUN Aug 6 2009...

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AMS 301 Lecture 7 Ning SUN Aug 6 2009

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Generating Functions
Integer Integer-solution solution-of of-an an-equation version equation version 1. The number of ways to select r objects with repetition from n different types of objects. 2. The number of ways to distribute r identical objects into n distinct boxes. 3. The number of nonnegative integer solutions to e 1 + e 2 +…+ e n = r . 12 = 4 + 3 + 1 + 4 AMS301, Summer 2009, Ning SUN

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Generating Function Suppose a r is the # of ways to select r objects in a certain procedure . Then g ( x ) is a generating function for a r if g ( x ) has the polynomial expansion ( 29 n n r r x a x a x a x a a x g + + + + + + = L L 2 2 1 0 A tool used for handling special constraints in selection and arrangement problems with repetition. AMS301, Summer 2009, Ning SUN
Binomial Theorem The coefficient of is . # of ways to select an r -subset from an n -set. g ( x ) is the generating function for . ( 29 n k n x n n x k n x n x n n x + + + + + + = + L L 2 2 1 0 1 ( 29 n x + 1 = r n a r r x r n AMS301, Summer 2009, Ning SUN

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Formal expansion of ( 29 3 1 x + ( 29 ( 29( 29( 29 3 2 3 3 3 1 1 1 11 1 1 1 11 111 1 1 1 1 x x x xxx xx x x x xx x x x x x x + + + = + + + + + + + = + + + = + ( 29 ( 29( 29( 29 1 1 1 0 1 1 1 0 1 0 0 1 1 1 0 0 1 0 1 0 0 0 0 0 1 0 1 0 1 0 3 1 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x + + + + + + + = + + + = + 3 2 1 e e e x x x 1 0 i e AMS301, Summer 2009, Ning SUN
Selection with repetition 4 types, at most two objects of each type Q: How many ways to select 5 objects? Q: What is the coefficient of ? 4 3 1 1 e e e e x x x x 2 0 i e 5 4 3 2 1 = + + + e e e e 5 x AMS301, Summer 2009, Ning SUN

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In the expansion, the set of all formal products will be sequences of the form: or Q: What is the coefficient of ? 2 1 0 2 1 0 2 1 0 2 1 0 x x x x x x x x x x x x 4 3 2 1 e e e e x x x x 2 0 i e 5 x 5 4 3 2 1 = + + + e e e e ( 29 4 2 1 x x + + Polynomial factor AMS301, Summer 2009, Ning SUN
Equivalent forms The coefficients of a generating function can be interpreted as the solutions to a certain selection-with-repetition or distribution-of- identical objects problem. Given a certain selection-with-repetition or distribution problem, we can build a generating function whose coefficients are the answers to this problem. AMS301, Summer 2009, Ning SUN

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Example 1 Q: Find the generating function for a r , the number of ways to select r balls from 3 black, 3 green, 3 blue, and 3 gold balls. 3 2 1 0 3 2 1 0 3 2 1 0 3 2 1 0 x x x x x x x x x x x x x x x x ( 29 = + + + = 4 3 2 1 ) ( x x x x g AMS301, Summer 2009, Ning SUN
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lec07 - AMS 301 Lecture 7 AMS Ning SUN Aug 6 2009...

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