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Discrete Math by R.S. Chang, Dept. CSIE, NDHU3 Chapter Introductory Examples Ex. 9.3 How many integer solutions are there for the equation
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Discrete Math by R.S. Chang, Dept. CSIE, NDHU 1
Chapter 9
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 2
Chapter 99.1 Introductory Examples
Ex. 9.1 Find the number of integer solutions toc c c c c c1 2 3 1 2 312 4 2 2 5 where , , .
c x x x x x c xc x x x x x c xc x x x x c x
x f x c x c x c x
f x generating function
14 5 6 7 8
1
22 3 4 5 6
2
32 3 4 5
3
1 2 3
: ( ): ( ): ( )
( ) ( ) ( ),
The coefficient of in ( ) =which is 14, is the answer.
( ) is called a for the distributions.
12
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 3
Chapter 99.1 Introductory Examples
Ex. 9.3 How many integer solutions are there for the equation
c c c c c ic
x x x x
f x x x x
g x x x x xx
x
ii
1 2 3 4
25
2 25 4
2 25 26 4
44
25 0 1 4
1
11
11
, , ?,
.
( )( )
For each the possibility can be described by1 + + Then the answer is the coefficient of in the generating function:
( ) = + + or
( ) = + +
2 25
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 4
Chapter 99.2 Definitions and Examples: Calculational Techniques
Def. 9.1 Let be a sequence of real numbers. The
function ( ) = is called the
generating function for the given sequence.
Ex. 9.4 For any Z
so (1 + ) is the generating function for the sequence
0
0
+
a a a
f x a a x a x a x
n xn n
xn
xnn
x
xn n n n
n
ii
i
n n
n
, , ,
, ( )
, , , , , , , , .
1 2
1 22
0
210 1 2
0 1 20 0 0
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 5
Chapter 99.2 Definitions and Examples: Calculational Techniques
Ex. 9.5 (a) For Z (1 -
So 1 - x is the generating function for , ,
b) If and | |< 1, 1 = ( - )( + + x
So 1-
is the generating function for 1,1,1, .
(c)
Consequently, 1
+ +
n+1
+ '2
n x x x x
x
n x x x x
xddx x
xx
ddx
x x x x x x
n n
n s
, ) ( )( ).
, , , .
( ).
( )( ) ( )( )
( )
1
1 13
22
2 3 2 3
1 1
111 1 0 0
1 1
11
11 1 1 1
11 1 2 3 4
( - ) is the generating function of 1,2,3, .
21 x
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 6
Chapter 99.2 Definitions and Examples: Calculational Techniques
Ex. 9.5 (continued)
And ( - )
is the generating function of 0,1,2,3, .
(d) Continuing from (c),
Hence +
( - ) generates 1 and
( + )
( - ) generates 0 1
2
2 2
x
xddx
x
x
x
xddx
x x x x x xx
xx x
x
1
1
1
10 2 3 1 2 3 4
1
12 3
1
12 3
2
2 3
2 3 2 2 2 2 3
32 2
32 2
( ) ( )( ) .
, , , ,
, , , , .
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 7
Chapter 99.2 Definitions and Examples: Calculational Techniques
Ex. 9.6 (a) generating function of 1,1,0,1,1, is 11 - x
The generating function for 1,1,1,3,1,1, is 11 - x
b) Find the generating function for 0,2,6,12,20,30,42,
Therefore, the generating function is
x
x
a a a aax x
x
x
x
x
x
2
3
02
12
22
32
42
3 2 3
2
0 0 0 2 1 1 6 2 2 12 3 320 4 4
1
1 1
2
1
(, , , ,
, .( )
( ) ( ) ( )
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 8
Chapter 99.2 Definitions and Examples: Calculational Techniques
Extension of binomial coefficientWith , Z and > 0, we have
If R, we use as the definition
of For example, if Z we have -
+
+
n r n rnr
nr n r
n n n n rr
n n n n n rr
nr
nnr
n n n n rr
n n n rr
n rr
r
r
!!( )!
( )( ) ( )!
.
( )( ) ( )!
. ,
( )( )( ) ( )!
( ) ( )( ) ( )!
( )
1 2 1
1 2 1
1 2 1 1 1 1
11
. . And for any real n, define
n0
1
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 9
Chapter 99.2 Definitions and Examples: Calculational Techniques
Ex. 9.7 For n Z
Ex. 9.8 Find the coefficient of x in (1 - 2x)
Ex. 9.10 Find the coefficient of x in f(x) = (x
The coefficient of x in
is -47
+
5 -7
15 2
7
, ( ) ( )
.
( ) ( ) ( ) , .
) .
( ) ( )( )
.
( )
1 11
75
2 17 5 1
532 14 784
11
1
1
0
0
5 5
3 4
2 2 4 8
4
4
xn r
rx
nr
x
x
f x x x x x
x
x
n r
r
r
r
r
( )1 1207
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 10
Chapter 99.2 Definitions and Examples: Calculational Techniques
Ex. 9.11 In how many ways can we select, with repetitions allowed,r objects from n distinct objects?
For each of the distinct objects, the geometric series+ + represents the possible choices for the
object. Considering all of the objects, the generating functionsis ( ) = ( + + and the required answer is the
coefficient of in ( ). ( ) = ( - )
So the answer is + -
2
-
nx x x
nf x x x x
x f x f x xni
x
n ii
xn r
r
n
r n i
i
i
i
1
1
1
1 1
3
2 3
0
0
) ,
. .
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 11
Chapter 99.2 Definitions and Examples: Calculational Techniques
Ex. 9.13 Verify that for all Z
Since (1 + ) by comparison of coefficients,
the coefficient of in ( + ) which is must equal the
coefficient of in and that is
+nn
nni
x x
x xnn
xn n
xnn
x
n nn
n nn
nn
i
n
n n
n n
n n
, .
[( ) ] ,
, ,
,
2
1
12
0 1
0 1 1
2
02 2
2
2
n nr
nn r0
. , With the result
follows.
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 12
Chapter 99.2 Definitions and Examples: Calculational Techniques
Ex. 9.14 Determine the coefficient of in ( - )( - )
partial fraction decomposition:
1
( - )( - ) or
1 = ( - ) By comparing coefficients, = , = -1, and = -1. Hence,
1
( - )( - )
2
xx x
x x
Ax
Bx
C
xA x B x x C x
A B C
x x x x x x
x
82
2
2
2 2
1
3 2
3 2 3 2 22 2 3 3
1
3 2
13
12
1
2
13
11 3
12
11 2
14
.
( )( )( ) ( ).
( ) ( / )
( / )
1
1 2 2( ( / ))x
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 13
Chapter 99.2 Definitions and Examples: Calculational Techniques
Ex. 9.15 Use generating functions to determine how many four-element subsets of S={1,2,3,...,15} contain no consecutiveintegers.Let { be one such subset with 1 Let
for 2 and Then with 0
and 2 Therefore, the answer is the coefficient of in ( ) = ( + + x
which is -58
1
=
2
a a a aa a a a c a c a a
i c a c c c
c c cx f x x x x x x
i i i
ii
1 2 3 41 2 3 4 1 1
5 41
51 5
2 3 414 2 2 3 3 6 5
8
15 1
4 15 14
1 1
1 495
, , , }. ,
, . ,
, , .) ( ) ( ) ,
( ) .
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 14
Chapter 99.2 Definitions and Examples: Calculational Techniques
Ex. 9.16 ( ) =( - )
generates 0,1,2,
and ( ) = ( + )
( - ) generates 0
Then ( ) = ( ) ( ) = where
2
=
f x x
xa a a
g x x x
xb b b
h x f x g x c x
c a b a b a b a b a b a bi k i i k ki i k i
kk
kk k k k k k k
ik
ik
ik
11
11 2
2
2 0 1 2
32 2
0 1 2
00 1 1 2 2 2 2 1 1 0
20
2 20
20
( , , , )
, , , ( , , , )
,
( ) ( )
21
22
1 2 16
12
20
30
2
2
0 1 2 0 1 2
k i i kk k
kk k k
k k a a a b b b
ik
ik ( ) ( )( )
( ) . , , , , , , . Convolution of and
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 15
Chapter 99.3 Partitions of Integers
Partition a positive integer n into positive summands and seeking the number of such partitions, without regard to order.For example, p(1)=1: 1 p(2)=2: 2=1+1 p(3)=3: 3=2+1=1+1+1 p(4)=5: 4=3+1=2+2=2+1+1=1+1+1+1 p(5)=7: 5=4+1=3+2=3+1+1=2+2+1=2+1+1+1 =1+1+1+1+1
We should like to obtain p(n) for a given n without having tolist all the partitions. We need a tool to keep track of the numbers of 1's, 2's, ..., n's that are used as summands for n.
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 16
Chapter 99.3 Partitions of Integers
keep track of 1's: 1 + +keep track of 2's: 1 + +
keep track of ' s: 1 + +For example, p(10) is the coefficient of in
( ) = ( + +
In general, ( ) = generate the sequence ( ), (1),
2
4
2k
10
2
x x xx x x
k x x xx
f x x x x x x
x x x x x
P xx
p p
k k
ii
ii
3
2 6
3
2 4 10
2 3 10 1
10
1
1 1 11
11
1
1
1
1
1
1
11
10
)( ) ( )
.
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 17
Chapter 99.3 Partitions of Integers
Ex. 9.18 Find the generating function for the number of ways anadvertising agent can purchase n minutes of air time if time slotsfor commercials come in blocks of 30, 60, or 120 seconds.
Let 30 seconds represent one time unit. Then the answer is thenumber of integer solutions to the equation + + =with 0 , , . The associated generating function is
( ) = ( + +
and the coefficient of is the answer.
2
a b c na b c
f x x x x x x x
x x xx n
2 4 2
1 1 11
11
1
1
1
2 4 4 8
2 42
)( )( )
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 18
Chapter 99.3 Partitions of Integers
Ex. 9.19 Find the generating function for pd(n), the number ofpartitions of a positive integer n into distinct summands.For any Z either is not used as a summand or it is. This can be accounted for by the polynomial 1 +Consequently, the generating function is
the generating function of partitioning into odd summands
+k kx
P x x x x x
xx
x
x
x
x
x
x x xx x x x x x P x
k
di
i
o
,.
( ) ( )( )( ) ( )
( )( )( ) ( )
1 1 1 1
11
1
1
1
1
1
1
11
1
11 1 1
2 3
12 4
2
6
3
8
4 32 3 6 5 10
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 19
Chapter 99.3 Partitions of Integers
Ex. 9.21 Partition into odd summands but each such odd summandsmust occur an odd number of times-or not at all.
f x x x x x x x x x
x k i
ik
( ) ( )( )(
) .( )( )
1 1 1
1
3 5 3 9 15 5 15
2 1 2 1
00
Ferrer's graph
14=4+3+3+2+1+1=6+4+3+1
The number of partitions ofan integer n into m summandsis equal to the number ofpartitions where m is the largest summands.
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 20
Chapter 99.4 The Exponential Generating Function
ordinary generating functions: selections (order is irrelevant)
( ) ( , ) ( , )!
( )
1
1 00
0 0
x C n r x P n rxr
x C n C nP n P n
n r
r
n r
r
n
n is the ordinary generating function of ( , ), ( ,1),But an exponential generating function for ( , ), ( ,1),
Def. 9.2 For a sequence of real numbers,
( ) =
is called the exponential generating function for the givensequence.
Ex. 9.22 is the ordinary g. f. of 1,1,12!
and
an exponential g. f. of 1,1,1, .
a a a
f x a a x ax
ax
axi
exi
i
i
i
xi
i
0 1 2
0 1 2
2
3
3
0
0
2 3
13
, , ,
! ! !
!,
!,
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 21
Chapter 99.4 The Exponential Generating Function
Ex. 9.23 In how many ways can four of the letters in ENGINEbe arranged?For the letter E and N, we use ( + + because thereare 0, 1, or 2 to arrange. For G and I, we use (1 + ).Consequetly, the exponential generating function (of
arrangement) is ( ) = + + And the answer
is the coefficient of !
in ( ).
12
12
1
4
2
2 22
4
xx
x
f x xx
x
xf x
!)
!( ) .
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 22
Chapter 99.4 The Exponential Generating Function
Ex. 9.24 e xx x
e xx x
e e x x
e ex
x x
x
x
x x
x x
12 3
12 3
21
2 4
2 3 5
2 3
2 3
2 4
3 5
! !
! !
! !
! !
Ex. 9.25 A ship carries 48 flags, 12 each of the colors red, white,blue, and black. Twelve of these flags are placed on a verticalpole in order to communicate a signal to other ships.
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 23
Chapter 99.4 The Exponential Generating Function
Ex. 9.25 (continued)(a) How many of these signals use an even number of blue flags andodd number of black flags?
( ) ( )! ! ! ! !
! !
a
the coefficient of in ( ) yields 411.
f x x x x x x x x
e e e e e e e e e
xi
x f x
xx x x x
x x x x
i
i
12
12 4 3 5
2 214
14
1
14
412
2 2 2 4 3 5
2 2 2 2 4
1
12
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 24
Chapter 99.4 The Exponential Generating Function
Ex. 9.25 (continued)(b) How many of these signals have at least three white flags orno white flags at all?
( ) ( )! ! !
!( )
!
! !
!
b
the coefficient of in ( ) yields 10,754,218.
g x xx x x
e e xx
e xex e x
i
xxi
x xi
x g x
x x x xx i
ii
i
i
i
12
13 4
2 24
32
3
12
2 3 3 4
3 24 3
2 3
0
0
2
012
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 25
Chapter 9Ex. 9.26 Assign 11 new employees to 4 subdivisions. Eachsubdivision will get at least one new employees..
Each subdivision can be selected from 1 time to 8 times.
Hence the e.g.f. is ( ) +!
The
coefficient of !
is what we want. It is easier to work with
+!
The coefficient of !
is
411
f x xx x x
x
x x x x e e e e
e x
ii
x x x x
x
i
i
2 3 8 4
11
2 3 8 44 4 3 2
11
11 11 11 11
0
4
2 3 8
11
2 3 81 4 6
4 111
4 3 6 2 4 1 14
4
! !.
! !
.
( ) ( ) ( ) ( ) ( )
the number of onto functions
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 26
Chapter 99.5 The Summation Operator
The prefix sum sequence of a sequence is Let ( ) be the generating
function of Then ( )-
Therefore, ( )-
is the g. f. of the prefix sum.sequence.
00
0
a a aa a a a a a f x
a a a f xx
a a x a x
x x a a a x a a a xf x
x
, , ,, , , .
, , , .
( ) ( )
1 20 1 0 1 2
1 2 0 1 22
20 0 1 0 1 2
21
1
1
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 27
Chapter 99.5 The Summation Operator
Ex. 9.27 Find a formula for 0
The g. f. of 0 is ( + )
( - )Therefore, the g. f. of 0 0 0 is
( + )
( - ) The coefficient of is what we wnat.
( + )
( - )
2
2
2 2 2
1 2
1 2 1
11 1 2
1
11
11
40
41
42
2 2 2
2 23
2 2 2
4
42 4
2 2
nx x
x
x x
xx
x x
xx x x
x x x x
n
.
, , , .
, , ,
.
( )( )
( )
Ansn n
n n n
n n: ( ) ( )
( )( )
41
142
1
1 2 16
1 2
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 28
Chapter 9Summaries (m objects, n containers)
Objects Containers Some Number Are Are Containers ofDistinct Distinct May Be Empty Distributions Yes Yes Yes nm Yes Yes No n!S(m,n) Yes No Yes S(m,1)+S(m,2)+...+S(m,n) Yes No No S(m,n) No Yes Yes No Yes No No No Yes (1) p(m), for n=m No No No (2) p(m,1)+p(m,2)+...+p(m,n), n<m p(m,n)
n mm
1n m n
m n
( ) 1
p(m.n):number of partitions of m into exactly n summands
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 29
Chapter 9
Exercise: P390: 6 P399: 18,20 P403: 9,10 P408: 6
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