4.6.2 Exponential generating functions The number of r-combinations of multiset

S={n1·a1,n2·a2,…, nk·ak} : C(r+k-1,r),

generating function:

kr

r

yyrrkC

)1(

1),1(

0

The number of r-permutation of set S={a1,a2,…, ak} :p(n,r),

generating function:

0

),(r

ryrnp

C(n,r)=p(n,r)/r!

n

r

n

r

rrn

r

xrnpxrnCx

0 0 !),(),()1(

n

r

r

r r

xa

0 !

Definition 2: The exponential generating function for the sequence a0,a1,…,an,…of real numbers is the infinite series

nn

r

rr xn

ax

axaax

r

axf

!!2!)( 22

100

0 !

)(

r

rax

r

axe

Theorem 4.17: Let S be the multiset {n1·a1,n2·a2,…,nk·ak} where n1,n2,…,nk are non-negative integers. Let br be the number of r-permutations of S. Then the exponential generating function g(x) for the sequence b1, b2,…, bk,… is given by

g(x)=gn1(x)·g n2

(x)·…·gnk(x) ， where for i=1,2,…,k,

gni(x)=1+x+x2/2!+…+xni/ni! .

(1)The coefficient of xr/r! in gn1(x)·g n2

(x)·…·gnk(x) is

0

2121!!!

!

i

km

rmmm kmmm

r

S of nspermutatio-r ofnumber theis

!!!

!)2(

02121

i

km

rmmm kmmm

r

Example: Let S={1·a1,1·a2,…,1·ak}. Determine the number r-permutations of S.

Solution: Let pr be the number r-permutations of S, and

k

r

r

k

r

r

k

r

rk

r

x

rn

n

xrnr

n

xrnCxxg

0

0

0

!)!(

!

)!(!

!

),()1()(

Example: Let S={·a1,·a2,…,·ak} ， Determine the number r-permutations of S.

Solution: Let pr be the number r-permutations of S,

gri(x)=(1+x+x2/2!+…+xr/r!+…) ， then

g(x)=(1+x+x2/2!+…+xr/r!+…)k=(ex)k=ekx

00 !!

)(

r

rr

r

r

r

xk

r

kx

Example ： Let S={2·x1,3·x2} ， Determine the number 4-permutations of S.

Let pr be the number r-permutations of S,

g(x)=(1+x+x2/2!)(1+x+x2/2!+x3/3!) Note: pr is coefficient of xr/r!.

Example ： Let S={2·x1,3·x2,4·x3}. Determine the number of 4-permutations of S so that each of the 3 types of objects occurs even times.

Solution: Let pr be the number r-permutations of S,

g(x)=(1+x2/2!)(1+x2/2!)(1+x2/2!+x4/4!)

Example: Let S={·a1,·a2, ·a3} ， Determine the number of r-permutations of S so that a3 occurs even times and a2 occurs at least one time.

Let pr be the number r-permutations of S, g(x)=(1+x+x2/2!+…+xr/r!+…)(x+x2/2!+…+xr/r! +

…) (1+x2/2!+x4/4!+…)=ex(ex-1)(ex+e-x)/2 =(e3x-e2x+ex-1)/2

1 !

)123(2

1

n

nnn

n

x

Example: Let S={·a1,·a2, ·a3} ， Determine the number of r-permutations of S so that a3 occurs odd times and a2 occurs at least one time.

Let pr be the number r-permutations of S, g(x)=(1+x+x2/2!+…+xr/r!+…)(x+x2/2!+…+xr/r! +

…) (x+x3/3!+x5/5!+…) =ex(ex-1)(ex-e-x)/2

4.7 Recurrence Relations P13, P100 Definition: A recurrence relation for the

sequence{an} is an equation that expresses an in terms of one or more of the previous terms of the sequence, namely, a0, a1, …, an-1, for all integers n with nn0, where n0 is a nonnegative integer.

A sequence is called a solution of a recurrence relation if its terms satisfy the recurrence relation.

Initial condition: the information about the beginning of the sequence.

Example(Fibonacci sequence): 13 世纪初意大利数学家 Fibonacci 研究过著名的

兔子繁殖数目问题 A young pair rabbits (one of each sex) is placed in

enclosure. A pair rabbits dose not breed until they are 2 months old, each pair of rabbits produces another pair each month. Find a recurrence relation for the number of pairs of rabbits in the enclosure after n months, assuming that no rabbits ever die.

Solution: Let Fn be the number of pairs of rabbits after n months,

(1)Born during month n (2)Present in month n-1 Fn=Fn-2+Fn-1 ， F1=F2=1

Example (The Tower of Hanoi): There are three pegs and n circular disks of increasing size on one peg, with the largest disk on the bottom. These disks are to be transferred, one at a time, onto another of the pegs, with the provision that at no time is one allowed to place a larger disk on top of a smaller one. The problem is to determine the number of moves necessary for the transfer.

Solution: Let h(n) denote the number of moves needed to solve the Tower of Hanoi problem with n disks. h(1)=1

(1)We must first transfer the top n-1 disks to a peg (2)Then we transfer the largest disk to the vacant peg (3)Lastly, we transfer the n-1 disks to the peg which

contains the largest disk. h(n)=2h(n-1)+1, h(1)=1

Using Characteristic roots to solve recurrence relations

Using Generating functions to solve recurrence relations

4.7.1 Using Characteristic roots to solve recurrence relations

Definition: A linear homogeneous recurrence relation of degree k with constant coefficients is a recurrence relation of the form

an=h1an-1+h2an-2+…+hkan-k, where hi are constants for all i=1,2,…,k,n≥k, and hk≠0.

Definition: A linear nonhomogeneous recurrence relation of degree k with constant coefficients is a recurrence relation of the form

an=h1an-1+h2an-2+…+hkan-k+f(n), where hi are constants for all i=1,2,…,k,n≥k, and hk≠0.

Definition: The equation xk-h1xk-1-h2xk-2-…-hk=0 is called the characteristic equation of the recurrence relation an=h1an-1+h2an-2+…+hkan-k. The solutions q1,q2,…,qk of this equation are called the characteristic root of the recurrence relation, where qi(i=1,2,…,k) is complex number.

Theorem 4.18: Suppose that the characteristic equation has k distinct roots q1,q2,…,qk. Then the general solution of the recurrence relation is

an=c1q1n+c2q2

n+…+ckqkn, where c1,c2,…ck are

constants.

Example: Solve the recurrence relation an=2an-1+2an-2 ， (n≥2) subject to the initial values a1=3 and a2=8. characteristic equation : x2-2x-2=0, roots: q1=1+31/2 ， q2=1-31/2。 the general solution of the recurrence relation is an=c1(1+31/2)n+c2(1-31/2)n, We want to determine c1 and c2 so that the initial values c1(1+31/2)+c2(1-31/2)=3, c1(1+31/2)2+c2(1-31/2)2=8

36

135

36

13521

cc

nnna )31(

36

135)31(

36

135

Theorem 4.19: Suppose that the characteristic equation has t distinct roots q1,q2,…,qt with multiplicities m1,m2,…,mt, respectively, so that mi≥1 for i=1,2,…,t and m1+m2+…+mt=k. Then the general solution of the recurrence relation is

t

i

m

j

ni

jijn

i

qnca1 1

1

where cij are constants for 1≤j≤mi and 1≤i≤t.

Example: Solve the recurrence relation an+an-1-3an-2-5an-3-2an-4=0,n≥4 subject to the initial values a0=1,a1=a2=0, and a3=2. characteristic equation x4+x3-3x2-5x-2=0, roots:-1,-1,-1,2 By Theorem 4.19 ： the general solution of the recurrence

relation is an=c11(-1)n+c12n(-1)n+c13n2(-1)n+c212n

We want to determine cij so that the initial values c11+c21=1 -c11-c12-c13+2c21=0 c11+2c12+4c13+4c21=0 -c11-3c12-9c13+8c21=2 c11=7/9,c12=-13/16,c13=1/16,c21=1/8 an=7/9(-1)n-(13/16)n(-1)n+(1/16)n2(-1)n+(1/8)2n

the general solution of the linear nonhomogeneous recurrence relation of degree k with constant coefficients is

an=a'n+a n*

a'n is the general solution of the linear homogeneous recurrence relation of degree k with constant coefficients an=h1an-1+h2an-2+…+hkan-k

a n*is a particular solution of the nonhomogeneous

linear recurrence relation with constant coefficients an=h1an-1+h2an-2+…+hkan-k+f(n)

Theorem 4.20: If {a n*} is a particular solution

of the nonhomogeneous linear recurrence relation with constant coefficients

an=h1an-1+h2an-2+…+hkan-k+f(n),

then every solution is of the form {a'n+a n*},

where {a n*} is a general solution of the

associated homogeneous recurrence relation an=h1an-1+h2an-2+…+hkan-k.

Key:a n*

(1)When f(n) is a polynomial in n of degree t, a n

*=P1nt+P2nt-1+…+Ptn+Pt+1

where P1,P2,…,Pt,Pt+1 are constant coefficients (2)When f(n) is a power function with constant

coefficient n, if is not a characteristic root of the associated homogeneous recurrence relation,

a n*= Pn ，

where P is a constant coefficient. if is a characteristic root of the associated

homogeneous recurrence relation with multiplicities m,

a n*= Pnmn ， where P is a constant coefficient.

Example: Find all solutions of the recurrence relation an+2an-1=n+1,n1, a0=2

Example: Find all solutions of the recurrence relation h(n)=2h(n-1)+1, n2, h(1)=1

Example: Find all solutions of the recurrence relation

an=an-1+7n,n1, a0=1

If let an*=P1n+P2,

P1n+P2-P1(n-1)-P2=7n

P1=7n

Contradiction let an*=P1n2+P2n

4.7.2 Using Generating functions to solve recurrence relations

0

)(n

nnxaxf

Example: Solve the recurrence relationan=an-1+9an-2-9an-3,n≥3subject to the initial values a0=0, a1=1, a2=2

Example: Solve the recurrence relation : an=an-1+9an-2-9an-3,n≥3

subject to the initial values a0=0, a1=1, a2=2

Solution: Let Generating functions of {an} be ： f(x)=a0+a1x+a2x2+…+anxn+… , then:

-xf(x)=-a0x-a1x2-a2x3…-anxn+1-…

-9x2f(x) = -9a0x2-9a1x3-9a2x4-…-9an-2xn-…

9x3f(x) = 9a0x3+9a1x4+…+9an-3xn+…

(1-x-9x2+9x3)f(x)=a0+(a1-a0)x+(a2-a1-9a0)x2+ (a3-a2-9a1+9a0)x3+…+(an-an-1-9an-2+9an-3)xn+…

a0=0,a1=1, a2=2 ， and when n≥3,an-an-1-9an-2+9an-3=0 ，(an=an-1+9an-2-9an-3)thus ：(1-x-9x2+9x3)f(x)=x+x2

f(x)=(x+x2)/(1-x-9x2+9x3)=(x+x2)/((1-x)(1+3x)(1-3x))

xxx 31

1

3

1

31

1

12

1

1

1

4

1

1/(1-x)=1+x+x2+…+xn+… ；1/(1+3x)=1-3x+32x2-…+(-1)n3nxn+…1/(1-3x)=1+3x+32x2+…+3nxn+… ； 1

13

4

3)1(

4

1

33

13)1(

12

1

4

1

nn

n

nnnna

Example: Find an explicit formula for the Fibonacci numbers,

Fn=Fn-2+Fn-1 ， F1=F2=1 。 Solution: Let Generating functions of {Fn} be ： f(x)=F0+F1x+F2x2+…+Fnxn+…,then ： -xf(x) =-F0x-F1x2-F2x3…-Fnxn+1-… -x2f(x) =-F0x2-F1x3-F2x4-…-Fn-2xn-… (1-x-x2)f(x)=F1x+(F2-F1)x2+(F3-F2-F1)x3+(F4-F3-

F2)x4+…+(Fn-Fn-1-Fn-2)xn+… F1=1, F2=1 ， and when n≥3,Fn-Fn-1-Fn-2=0 ， (Fn=Fn-1+Fn-2) thus ： (1-x-x2)f(x)=x f(x)=x/(1-x-x2)

n

n

nn x))()((5

1

02

512

51

))()((5

12

512

51 nnnF

0)(,1|| 251

251 lim

n

n

hence

2

51

)(

1

251

51

n

n

nn

F

F

F

618.051

21

n

n

F

FThus

Fn-10.618Fn 。golden section 黄金分割。

xx

xx

xx

x

251

251

251

51

251

51

251

251

1

1

1

1

5

1

11

2

1

))((

Exercise P104 18,20,23.Note: By Characteristic roots, solve recurrence relations 23; By Generating functions, solve recurrence relations 18,20.

1.Determine the number of n digit numbers with all digits at least 4, such that 4 and 6 each occur an even number of times, and 5 and 7 each occur at least once, there being no restriction on the digits 8 and 9.

2.a)Find a recurrence relation for the number of ways to climb n stairs if the person climbing the stairs can take one stair or two stairs at a time. b) What are the initial conditions?

3.a) Find a recurrence relation for the number of ternary strings that do not contain two consecutive 0s. b) What are the initial conditions?