Chapter 9 Generating functions

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Chapter 9 Generating functions

• Yen-Liang Chen
• Dept of Information Management
• National Central University

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9.1. Introductory examples

• Ex 9.1.
• 12 oranges for three children, Grace, Mary, and
  Frank.
• Grace gets at least four, and Mary and Frank gets
  at least two, but Frank gets no more than five.
• (x4 x5 x6 x7 x8) (x2 x3x4 x5 x6)(x2
  x3x4 x5)
• The coefficient of x12 is the solution.

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Ex 9.2

• Four kinds of jelly beans, Red, Green, White,
  Black
• In how many ways can we select 24 jelly beans so
  that we have an even number of white beans and at
  least six black ones?
• Red (green) 1 x1 x2. x23 x24
• White 1 x2 x4. x22 x24
• Black x6 x7. x23 x24
• f(x)(1 x1 x2. x23 x24)(1 x2 x4. x22
  x24)(x6 x7. x23 x24)
• The coefficient of x24 is the solution.

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Ex 9.3.

• How many nonnegative integer solutions are there
  for c1c2c3c425?
• f(x)(1 x1 x2. x24 x25)4
• The coefficient of x25 is the solution.

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9.2. Definition and examples calculational
techniques

• Ex 9.4. (1x)n is the generating function for the
  sequence C(n, 0), C(n, 1),, C(n, n), 0,0,0

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Ex 9.5

• (1-xn1)/(1-x) is the generating function for the
  sequence 1,1,1,,1, 0, 0,0, where the first n1
  terms are 1.
• 1/(1-x) is the generating function for the
  sequence 1,1,1,,1,.
• 1/(1-x)2 is the generating function for the
  sequence 1,2,3,4,.
• x/(1-x)2 is the generating function for the
  sequence 0,1,2,3,.
• (x1)/(1-x)3 is the generating function for the
  sequence 12,22,32,42,.
• x(x1)/(1-x)3 is the generating function for the
  sequence 02,12,22,32,42,.

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Ex 9.6

• 1/(1-ax) is the generating function for the
  sequence a0,a1,a2,a3,.
• Let f(x)1/(1-x). Then g(x)f(x)-x2 is the
  generating function for the sequence 1,1,0, 1,
  1,,.
• Let aii2i for i?0. Then its generating function
  is x(x1)/(1-x)3x/(1-x)22x/(1-x)3

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Define C(n, r) for n?R

• Since when n?Z, we have
• So for n ?R we define
• For example, if n is positive, we have

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Tayors and Maclaurins Series

• f(x)f(c) (x-c) f?(c) (x-c)2f(c)/2!
  (x-c)3f(3)(c)/3! (x-c)4f(4)(c)/4! ..
  (x-c)nf(n)(c)/n!
• Let c0. Then we have
• f(x)f(0) (x) f?(0) (x)2f(0)/2!
  (x)3f(3)(0)/3! (x)4f(4)(0)/4! ..
  (x)nf(n)(0)/n!

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Ex 9.7
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Ex 9.9
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Ex 9.10.

• Determine the coefficient of x15 in
  f(x)(x2x3x4)4.
• (x2x3x4)x2(1xx2)x2/(1-x)
• f(x)(x2/(1-x))4x8/(1-x)4
• Hence the solution is the coefficient of x7 in
  (1-x)-4, which is C(-4, 7)(-1)7C(10, 7).

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Ex 9.11

• In how many ways can we select, with repetition
  allowed, r objects from n distinct objects?
• Consider f(x)(1xx2)n
• (1/(1-x))n1/(1-x)n
• The coefficient of xr is C(nr-1, r)
• The answer is the coefficient of xr in f(x).

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Ex 9.12

• x/(1-x) x1 x2x3x4.
• The coefficient of xj in (x1x2x3x4.)i is the
  number of ways that we form the integer j by i
  summands.
• The number of ways to form an integer n is the
  coefficient of xn in the following generating
  function.

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Ex 9.14.

• In how many ways can a police captain distribute
  24 rifle shells to four police officers, so that
  each officer gets at least three shells but not
  more than eight.
• f(x) (x3x4 x5 x6x7x8)4
• x12(1xx2x3x4x5)4
• x12(1-x6)/(1-x)4
• the answer is the coefficient of x12 in
  (1-x6)4(1-x)-4

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Ex 9.17.

• How many four element subsets of S1, 2,, 15
  contains no consecutive integers?
• 1, 3, 7, 10 ? 0, 2, 4, 3, 5
• there exists a one-to-one correspondence between
  the four-element subsets to be counted and the
  integer solutions to c1c2c3c4c514 where
  0?c1, c5 and 2?c2, c3, c4.
• The answer is the coefficient of x14 in the
  following formula
• f(x)(1xx2x3)(x2x3x4)3(1xx2x3)x6(1
  -x)-5

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Ex 9.17.

• 1, 3, 7, 10 ? 0, 1, 3, 2, 5
• there exists a one-to-one correspondence between
  the four-element subsets to be counted and the
  integer solutions to c1c2c3c4c511 where
  0?c1, c5 and 1?c2, c3, c4.
• The answer is the coefficient of x14 in the
  following formula
• f(x)(1xx2x3)(xx2x3x4)3(1xx2x3)x3(
  1-x)-5

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Ex 9.18.

• Brianna takes an examination until she passes it.
  Suppose in each test the probability of failure
  is 0.8, and of success is 0.2.
• Let Y denote the number of times Brianna expects
  to take the exam before she passes it.
• Please compute E(Y) and E(Y2).

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compute E(Y)
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compute E(Y2).
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the convolution of sequences

• Ex 9.19. Let f(x)x/(1-x)201x2x23x3, where
  aii
• g(x)x(x1)/(1-x)3012x22x232x3, where bii2
• h(x)f(x)g(x)c0c1xc2x2c3x3
• cka0bka1bk-1a2bk-2ak-2b2ak-1b1akb0
• ckthe sequence c is the convolution of sequences
  a and b

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9.3. Partition of integers

• p(x) is the number of partitions for x.
• For n, the number of 1s is 0 or 1 or 2 or 3.
  The power series is 1xx2x3x4.
• For n, the number of 2s can be kept tracked by
  the power series 1x2x4x6x8.
• For n, the number of 3s can be kept tracked by
  the power series 1x3x6x9x12.
• f(x)(1xx2x3x4)(1x2x4x6x8x10)
  (1x3x6x9) (1x10)
• 1/(1-x)?1/(1-x2)? 1/(1-x3) ??
  1/(1-x10)
• At last, we have the following series for p(n) by
  the coefficient of xn

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Ex 9.21

• Find the number of ways an advertising agent can
  purchase n minutes if the time slots come in
  blocks of 30, 60, 120 seconds.
• Let 30 seconds represent one time unit.
• a2b4c2n
• f(x) (1xx2x3x4) (1x2x4x6x8)(
  1x4x8x12)
• 1/(1-x) ? 1/(1-x2) ? 1/(1-x4).
• The coefficient of x2n is the answer to the
  problem.

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Examples

• Ex 9.22. pd(n) is the number of partitions of a
  positive integer n into distinct summands.
• Pd(x)(1x)(1x2)(1x3)..
• Ex 9.23. po(n) is the number of partitions of a
  positive integer n into odd summands.
• Po(x) (1xx2x3x4) (1x3x6x9x12)(
  1x5x10x15)
• Po(x)1/(1-x) ? 1/(1-x3) ? 1/(1-x5) ? 1/(1-x7) ?
  ...
• Pd(x) Po(x)

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Ex 9.24.

• poo(n) is the number of partitions of a positive
  integer n into odd summands and such summands
  must occur an odd number of times.
• Poo(x) (1xx3x5x7) (1x3x9x15)(
  1x5x15x25)

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9.4. The exponential generating function
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Ex 9.26

• In how many ways can four of the letters in
  ENGINE be arranged?
• f(x)1x(x2/2!)21x2, and the answer is the
  coefficient of x4/4!.

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important series
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Ex 9.28.

• We have 48 flags, 12 each of the colors red,
  white, blue and black. Twelve flags are placed on
  a vertical pole to show signal.
• How many of these use an even number of blue
  flags and an odd number of black flags?
• l

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Ex 9.28

• how many of these use at least three white flags
  or no white flag at all?

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Ex 9.29.

• A company hires 11 new employees, and they will
  be assigned to four different departments, A, B,
  C, D. Each department has at least one new
  employee. In how many ways can these assignments
  be done?

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9.5. The summation operator

• Let f(x)a0a1xa2x2a3x3. Then f(x)/(1-x)
  generate the sequence of a0, a0a1, a0a1a2,
  a0a1a2a3, So we refer to 1/(1-x) as the
  summation operator.

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Ex 9.30.

• 1/(1-x) is the generating function for the
  sequence 1, 1, 1, 1, 1,
• 1/(1-x)? 1/(1-x) is the generating function
  for the sequence 1,2,3,4,5,
• xx2 is the generating function for the sequence
  0, 1, 1, 0, 0, 0,
• (xx2) /(1-x) is the generating function for the
  sequence 0, 1, 2, 2, 2, 2,
• (xx2) /(1-x)2 is the generating function for the
  sequence 0, 1, 3, 5, 7, 9, 11,
• (xx2) /(1-x)3 is the generating function for the
  sequence 0, 1, 4, 9, 16, 25, 36,

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Ex 9.31.

• g(x) 1/(1-x)1xx2x3x4
• q(x)dg(x)/dx1/(1-x)212x3x24x3.
• r(x)xq(x)x/(1-x)2 x2x23x34x4.
• xdr(x)/dx(1x)/(1-x)3 x22x232x342x4.
• x(1x)/(1-x)4 x(1222)x2(122232)x3(122232
  42)x4.