Generating Functions

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Generating Functions
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Learning Objectives

• Understand what are generating functions.
• Understand how generating functions can be used
  to solve advanced counting problems.
• Understand how generating functions can be used
  to solve recurrence relations.

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Generating Functions

• Generating functions are useful for manipulating
  sequences and therefore for solving counting
  problems.
• Definition Let S a0, a1, a2, be an
  (infinite) sequence of real numbers. Then the
  generating function G(x), of S is the series

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Generating Functions

• ExamplesLet S 1, 1, 1, then
• Let S 0, 1, 2, 3, then

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Generating Functions

• ExamplesLet S 1, 1, 1 thenpad with an
  infinite number of zeros 1, 1, 1, 0, 0,
  a closed form for G.

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Generating Functions

• Extended binomial theorem The binomial theorem
  can be extended when n is not an
  integer the sum does not terminate.Let
  x be a real number with x lt 1 and let u be a
  real number. Then

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Generating Functions

• Example Calculate 1/(1-3x2)
  where a1, b(-3x2), and n -1This
  expression is the generating function for the
  sequenceS1, 0, 3, 0, 32, 0, 33, 0,

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Generating Functions

• Manipulation of generating functionsLet
• Then
• Sum
• Product

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Generating Functions

• Example Calculate 1/(1-x)2

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Generating Functions

• Generating functions are useful for solving
  counting problems, and also recurrence relations.
• ExampleSuppose we wish to count the integer
  solutions toa b 10 where a and b are
  constrained1 a 6 and 3 b 9there are
  6 possible solutions1 92 83 74 65
  56 4

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Generating Functions

• The sequences A 0, 1, 1, 1, 1, 1, 1, 0, 0, 0,
  where ak 1 if k is a possible value of a and
  B 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0,
  where bk 1 if k is a possible value of b
  then the product of the two generating functions
  for A and B will have the total possible
  solutions to a b 10as a coefficient of x10
  in (x x2 x3 x4 x5 x6) (x3 x4 x5
  x6 x7 x8 x9)

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Generating Functions

• Examplein how many ways can 8 identical cookies
  be distributed among 3 distinct children if each
  child receives at least 2 cookies and no more
  than 4 cookies?Each child receives x2 x3 x4
  cookiesThere are 3 children, therefore the
  generating function is ( x2 x3 x4 )3Since
  we are interested in 8 cookies, the term of x8 is
  what we are looking for. Its coefficient is 6,
  therefore the number of ways to distribute the
  cookies is 6.

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Generating Functions