4.6.2 Exponential generating functions

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• 4.6.2 Exponential generating functions
• The number of r-combinations of multiset
  Sn1a1,n2a2,, nkak C(rk-1,r),
• generating function

The number of r-permutation of set Sa1,a2,,
ak p(n,r), generating function
2

• C(n,r)p(n,r)/r!

Definition 2 The exponential generating function
for the sequence a0,a1,,an,of real numbers is
the infinite series
3

• Theorem 4.17 Let S be the multiset
  n1a1,n2a2,,nkak 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 i1,2,,k,
• gni(x)1xx2/2!xni/ni! .
• (1)The coefficient of xr/r! in gn1(x)g
  n2(x)gnk(x) is

4

• Example Let S1a1,1a2,,1ak. Determine the
  number r-permutations of S.
• Solution Let pr be the number r-permutations of
  S, and

5

• 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)(1xx2/2!xr/r!),then
• g(x)(1xx2/2!xr/r!)k(ex)kekx

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• ExampleLet S2x1,3x2,Determine the number
  4-permutations of S.
• Let pr be the number r-permutations of S,
• g(x)(1xx2/2!)(1xx2/2!x3/3!)
• Note pr is coefficient of xr/r!.
• ExampleLet S2x1,3x2,4x3. 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)(1x2/2!)(1x2/2!)(1x2/2!x4/4!)

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• 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)(1xx2/2!xr/r!)(xx2/2!xr/r! )
  (1x2/2!x4/4!)ex(ex-1)(exe-x)/2
• (e3x-e2xex-1)/2

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• 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)(1xx2/2!xr/r!)(xx2/2!xr/r! )
  (xx3/3!x5/5!)
• ex(ex-1)(ex-e-x)/2

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4.7 Recurrence Relations

• P13, P100
• Definition A recurrence relation for the
  sequencean 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 n?n0, 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.

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• Example(Fibonacci sequence)
• 13 ????????? Fibonacci ??????????????
• 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
• FnFn-2Fn-1,F1F21

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• 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

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• Using Characteristic roots to solve recurrence
  relations
• Using Generating functions to solve recurrence
  relations

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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
• anh1an-1h2an-2hkan-k, where hi are constants
  for all i1,2,,k,nk, and hk?0.
• Definition A linear nonhomogeneous recurrence
  relation of degree k with constant coefficients
  is a recurrence relation of the form
• anh1an-1h2an-2hkan-kf(n), where hi are
  constants for all i1,2,,k,nk, and hk?0.

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• Definition The equation xk-h1xk-1-h2xk-2--hk0
  is called the characteristic equation of the
  recurrence relation anh1an-1h2an-2hkan-k.
  The solutions q1,q2,,qk of this equation are
  called the characteristic root of the recurrence
  relation, where qi(i1,2,,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
• anc1q1nc2q2nckqkn, where c1,c2,ck are
  constants.

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• Example Solve the recurrence relation
• an2an-12an-2,(n2)
• subject to the initial values a13 and a28.
• characteristic equation
• x2-2x-20,
• roots
• q1131/2,q21-31/2?
• the general solution of the recurrence relation
  is
• anc1(131/2)nc2(1-31/2)n,
• We want to determine c1 and c2 so that the
  initial values
• c1(131/2)c2(1-31/2)3,
• c1(131/2)2c2(1-31/2)28

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• Theorem 4.19 Suppose that the characteristic
  equation has t distinct roots q1,q2,,qt with
  multiplicities m1,m2,,mt, respectively, so that
  mi1 for i1,2,,t and m1m2mtk. Then the
  general solution of the recurrence relation is

where cij are constants for 1jmi and 1it.
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• Example Solve the recurrence relation
• anan-1-3an-2-5an-3-2an-40,n4
• subject to the initial values a01,a1a20, and
  a32.
• characteristic equation
• x4x3-3x2-5x-20,
• roots-1,-1,-1,2
• By Theorem 4.19the general solution of the
  recurrence relation is
• anc11(-1)nc12n(-1)nc13n2(-1)nc212n
• We want to determine cij so that the initial
  values
• c11c211
• -c11-c12-c132c210
• c112c124c134c210
• -c11-3c12-9c138c212
• c117/9,c12-13/16,c131/16,c211/8
• an7/9(-1)n-(13/16)n(-1)n(1/16)n2(-1)n(1/8)2n

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• the general solution of the linear nonhomogeneous
  recurrence relation of degree k with constant
  coefficients is
• ana'na n
• a'n is the general solution of the linear
  homogeneous recurrence relation of degree k with
  constant coefficients anh1an-1h2an-2hkan-k
• a nis a particular solution of the
  nonhomogeneous linear recurrence relation with
  constant coefficients
• anh1an-1h2an-2hkan-kf(n)

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• Theorem 4.20 If a n is a particular solution
  of the nonhomogeneous linear recurrence relation
  with constant coefficients
• anh1an-1h2an-2hkan-kf(n),
• then every solution is of the form a'na n,
  where a n is a general solution of the
  associated homogeneous recurrence relation
  anh1an-1h2an-2hkan-k.
• Keya n

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• (1)When f(n) is a polynomial in n of degree t,
• a nP1ntP2nt-1PtnPt1
• where P1,P2,,Pt,Pt1 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 P??n ,
• where P is a constant coefficient.
• if ? is a characteristic root of the associated
  homogeneous recurrence relation with
  multiplicities m,
• a n P?nm??n ,where P is a constant coefficient.
• Example Find all solutions of the recurrence
  relation an2an-1n1,n?1, a02

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• Example Find all solutions of the recurrence
  relation h(n)2h(n-1)1, n?2, h(1)1
• Example Find all solutions of the recurrence
  relation
• anan-17n,n?1, a01
• If let anP1nP2,
• P1nP2-P1(n-1)-P27n
• P17n
• Contradiction
• let anP1n2P2n

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4.7.2 Using Generating functions to solve
recurrence relations
Example Solve the recurrence relation anan-19an
-2-9an-3,n3 subject to the initial values a00,
a11, a22
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• Example Solve the recurrence relation
• anan-19an-2-9an-3,n3
• subject to the initial values a00, a11, a22
• Solution Let Generating functions of an be
• f(x)a0a1xa2x2anxn , then
• -xf(x) -a0x-a1x2-a2x3-anxn1-
• -9x2f(x) -9a0x2-9a1x3-9a2x4--9an-2xn-
• 9x3f(x) 9a0x39a1x49an-3xn
• (1-x-9x29x3)f(x)a0(a1-a0)x(a2-a1-9a0)x2
  (a3-a2-9a19a0)x3(an-an-1-9an-29an-3)xn

a00,a11, a22,and when n3,an-an-1-9an-29an-30
, (anan-19an-2-9an-3) thus (1-x-9x29x3)f(x)x
x2 f(x)(xx2)/(1-x-9x29x3) (xx2)/((1-x)(13x)(
1-3x))
1/(1-x)1xx2xn 1/(13x)1-3x32x2-(-1)n3
nxn 1/(1-3x)13x32x23nxn
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• Example Find an explicit formula for the
  Fibonacci numbers,
• FnFn-2Fn-1,
• F1F21?
• Solution Let Generating functions of Fn be
• f(x)F0F1xF2x2Fnxn,then
• -xf(x) -F0x-F1x2-F2x3-Fnxn1-
• -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
• F11, F21,and when n3,Fn-Fn-1-Fn-20,
• (FnFn-1Fn-2)
• thus
• (1-x-x2)f(x)x
• f(x)x/(1-x-x2)

Fn-1?0.618Fn? golden section?????
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• 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?