Chapter 9 Generating functionsYen-Liang Chen

Dept of Information Management

National Central University

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+ x3+x4+ x5+ x6)(x2+ x3+x4+ x5)

The coefficient of x12 is the solution.

Ex 9.2 Four kinds of jelly beans, Red, Green, White, Black In how many ways can we select 24 jelly beans so th

at 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.

Ex 9.3. How many nonnegative integer solutions are t

here for c1+c2+c3+c4=25? f(x)=(1+ x1+ x2+….+ x24+ x25)4

The coefficient of x25 is the solution.

9.2. Definition and examples: calculational techniques

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

Ex 9.5 (1-xn+1)/(1-x) is the generating function for the sequence

1,1,1,…,1, 0, 0,0…, where the first n+1 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,

…. (x+1)/(1-x)3 is the generating function for the sequence

12,22,32,42,…. x(x+1)/(1-x)3 is the generating function for the sequence

02,12,22,32,42,….

Ex 9.6 1/(1-ax) is the generating function for the seq

uence a0,a1,a2,a3,…. Let f(x)=1/(1-x). Then g(x)=f(x)-x2 is the gen

erating function for the sequence 1,1,0, 1, 1,…,….

Let ai=i2+i for i0. Then its generating function is [x(x+1)/(1-x)3]+[x/(1-x)2]=2x/(1-x)3

Define C(n, r) for nR Since when nZ+, we have

So for n R we define

For example, if n is positive, we have

!

)1)...(2)(1(

)!(!

!

r

rnnnn

rnr

n

r

n

!

)1)...(2)(1(

r

rnnnn

r

n

r

rn

rn

rn

r

rnnn

r

rnnnn

r

n

rr

r

1)1(

!)!1(

)!1()1(

!

)1)...(1)(()1(

!

)1)...(2)(1)((

Tayor’s and Maclaurin’s 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 c=0. 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!

Ex 9.7

Ex 9.9

Ex 9.10. Determine the coefficient of x15 in f(x)=(x2+x3

+x4+…)4. (x2+x3+x4+…)=x2(1+x+x2+…)=x2/(1-x) f(x)=(x2/(1-x))4=x8/(1-x)4

Hence the solution is the coefficient of x7 in (1-x)-4, which is C(-4, 7)(-1)7=C(10, 7).

Ex 9.11 In how many ways can we select, with repetiti

on allowed, r objects from n distinct objects? Consider f(x)=(1+x+x2+…)n

(1/(1-x))n=1/(1-x)n

The coefficient of xr is C(n+r-1, r) The answer is the coefficient of xr in f(x).

Ex 9.12 x/(1-x)= x1+ x2+x3+x4+…. The coefficient of xj in (x1+x2+x3+x4+….)i is the number of w

ays 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.

1 1

321 ))1/((...)(i i

ii xxxxx

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)= (x3+x4+ x5+ x6+x7+x8)4

=x12(1+x+x2+x3+x4+x5)4

=x12[(1-x6)/(1-x)]4

the answer is the coefficient of x12 in (1-x6)4(1-x)-4

Ex 9.17. How many four element subsets of S={1, 2,…, 15}

contains no consecutive integers? {1, 3, 7, 10} 0, 2, 4, 3, 5 there exists a one-to-one correspondence between t

he four-element subsets to be counted and the integer solutions to c1+c2+c3+c4+c5=14 where 0c1, c5 and 2c2, c3, c4.

The answer is the coefficient of x14 in the following formulaf(x)=(1+x+x2+x3+…)(x2+x3+x4+…)3(1+x+x2+x3+…)=x6(1-x)-5

Ex 9.17. {1, 3, 7, 10} 0, 1, 3, 2, 5 there exists a one-to-one correspondence between t

he four-element subsets to be counted and the integer solutions to c1+c2+c3+c4+c5=11 where 0c1, c5 and 1c2, c3, c4.

The answer is the coefficient of x14 in the following formula

f(x)=(1+x+x2+x3+…)(x+x2+x3+x4+…)3(1+x+x2+x3+…)=x3(1-x)-5

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).

compute E(Y)

compute E(Y2).

the convolution of sequences Ex 9.19. Let f(x)=x/(1-x)2=0+1x+2x2+3x3+…,

where ai=i g(x)=x(x+1)/(1-x)3=0+12x+22x2+32x3+…, whe

re bi=i2

h(x)=f(x)g(x)=c0+c1x+c2x2+c3x3+… ck=a0bk+a1bk-1+a2bk-2+…+ak-2b2+ak-1b1+akb0

ck=the sequence c is the convolution of sequences a and b

9.3. Partition of integers p(x) is the number of partitions for x. For n, the number of 1’s is 0 or 1 or 2 or 3…. The power series is 1+x+x2

+x3+x4+…. For n, the number of 2’s can be kept tracked by the power series 1+x2+x4

+x6+x8+…. For n, the number of 3’s can be kept tracked by the power series 1+x3+x6

+x9+x12+…. f(x)=(1+x+x2+x3+x4+…)(1+x2+x4+x6+x8+x10+…) (1+x3+x6+x9+…) …

(1+x10+…) =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

)]1/(1[1i

i x

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. a+2b+4c=2n f(x)= (1+x+x2+x3+x4+…) (1+x2+x4+x6+x8+…)( 1+x

4+x8+x12+…)

=1/(1-x) 1/(1-x2) 1/(1-x4). The coefficient of x2n is the answer to the problem.

Examples Ex 9.22. pd(n) is the number of partitions of a pos

itive integer n into distinct summands. Pd(x)=(1+x)(1+x2)(1+x3)..…

Ex 9.23. po(n) is the number of partitions of a positive integer n into odd summands. Po(x)= (1+x+x2+x3+x4+…) (1+x3+x6+x9+x12+…)( 1+

x5+x10+x15+…)… Po(x)=1/(1-x) 1/(1-x3) 1/(1-x5) 1/(1-x7) ... Pd(x)= Po(x)

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)= (1+x+x3+x5+x7+…) (1+x3+x9+x15+…)( 1+x5+x15+x25+…)…

9.4. The exponential generating function

Ex 9.26 In how many ways can four of the letters in ENGIN

E be arranged? f(x)=[1+x+(x2/2!)]2[1+x]2, and the answer is the coef

ficient of x4/4!.

important series

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

Ex 9.28 how many of these use at least three white flags or

no white flag at all?

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?

9.5. The summation operator Let f(x)=a0+a1x+a2x2+a3x3+…. Then f(x)/(1-x) gener

ate the sequence of a0, a0+a1, a0+a1+a2, a0+a1+a2+a3,… So we refer to 1/(1-x) as the summation operator.

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,… x+x2 is the generating function for the sequence 0, 1, 1, 0, 0,

0,… (x+x2) /(1-x) is the generating function for the sequence 0, 1,

2, 2, 2, 2, … (x+x2) /(1-x)2 is the generating function for the sequence 0, 1,

3, 5, 7, 9, 11, … (x+x2) /(1-x)3 is the generating function for the sequence 0, 1,

4, 9, 16, 25, 36, …

Ex 9.31. g(x)= 1/(1-x)=1+x+x2+x3+x4+… q(x)=dg(x)/dx=1/(1-x)2=1+2x+3x2+4x3+.… r(x)=xq(x)=x/(1-x)2 = x+2x2+3x3+4x4+.… xdr(x)/dx=(1+x)/(1-x)3= x+22x2+32x3+42x4+. x(1+x)/(1-x)4 = x+(12+22)x2+(12+22+32)x3+(12

+22+32+42)x4+.…