This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: AMS 301 Lecture 7 AMS 301 Lecture 7 Ning SUN Aug 6 2009 Generating Functions Generating Functions Integer Integersolution solutionof ofan anequation 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 3 AMS301, Summer 2009, Ning SUN Generating Function 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 A tool used for handling special constraints in selection and arrangement problems with repetition. 4 AMS301, Summer 2009, Ning SUN • Binomial Theorem • The coefficient of is . • # of ways to select an rsubset from an nset. • 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 1 ( 29 n x + 1 = r n a r r x r n 5 AMS301, Summer 2009, Ning SUN Formal expansion of 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 1 1 1 1 1 1 1 1 1 1 1 1 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 ≤ ≤ i e 6 AMS301, Summer 2009, Ning SUN Selection with repetition 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 ≤ ≤ i e 5 4 3 2 1 = + + + e e e e 5 x 7 AMS301, Summer 2009, Ning SUN In the expansion, the set of all formal products will be sequences of the form: or Q: What is the coefficient of ? ⋅ ⋅ ⋅ 2 1 2 1 2 1 2 1 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 ≤ ≤ i e 5 x 5 4 3 2 1 = + + + e e e e ( 29 4 2 1 x x + + Polynomial factor 8 AMS301, Summer 2009, Ning SUN Equivalent forms Equivalent forms • The coefficients of a generating function can be interpreted as the solutions to a certain selectionwithrepetition or distributionof identical objects problem. • Given a certain selectionwithrepetition or distribution problem, we can build a generating function whose coefficients are the answers to this problem....
View
Full
Document
 Spring '08
 ARKIN
 Sun Microsystems, Coef, SunOS, Sun2, Sun3, Ning SUN

Click to edit the document details