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MT2504 Combinatorics and Probability: Chapter 3

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  • August 3, 2022
  • 7
  • 2020/2021
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WinterBerry
CHAPTER 3 :




RECURSION AND GENERATING FUNCTIONS

RECURSIVE SEQUENCES

E. g (t - l ) a EIR



}
' "
f- (2) f- )
Cny
f- ( o ) -
- I
flo)
-
- I
,
fcc ) - a
,
= a =
a
,
. . .


,


f- ( htt) = a
f- ( )u
induction to show
use




E -



g ( C 27 -




} ! induction
I
f- Co) flu) n ← use
-
-
-
-




flute) = ( htt ) f- Ca)




E. g ( I -3 )
" ( rt "t )
'



Express flu) =

recursively .




via induction




)
"
S" " " t" "
"" = ° " "" "t " "" = " "" + " "" t '" " " "d to " th "
P"
=
"

"




f- ( c ) = l
flu ) =
02 ✓

tf (
' '

fcc) (n ( htt) ( Zntl ))
'
2
f Cz ) = 5 = t
flu ) t ti ) =
n t ( ht D 2n2+7nt6 = 2n2t4nt3nt 6
"

By t ( ntl ) (
#
f- (z )
= 14 =
f- (2) t 3 # H . =
n ( zntl) t G ( ht i ) =
2n (n t 2) t3(nt2 )

I ( htt ) ( 2n2 7- 6 ] ( znt 3) ( ntz)
=
=
+ n t


=
I ( atl ) ( ht 2) ( 2h t 3)

=
( htt ) ( ( ht c) t 1) ( 2 ( nti ) ti )
6 So the result holds for all n> o .




E.
g.
( 1.4 )

we define a recursive function fln ,
r ) for her > o and n > O as follows :




f- (n ,
o ) = f- ( n ,
n ) = I


f- ( n
,
r ) =
f (n -
I
,
r -
l ) t f- (n -
I
,
r )


( I It ( I ) )
"

Pascals identity ( ( F ) satisfies the conditions and boundary conditions f
:
same as .




So far r )
-

-

(7)

,×4
,




In :') F
vi. it
'
'

+
,


it: i!
" """ "
: innit ! n'÷
'
! inn
'
'
'
+

in .ir .net
-
= -




.
. r ! (n -
r )!



= r (n -
i )!
! (n
+ (n -
r ) (n -
i )! = (n -
i ) ! ( Yt

! (n )!
n -


A =




! (n
n !

)!
( =
( Y ))
r
-
r )! r -
r
r
-
r

, FIBONACCI NUMBERS
Definition :( 2. c)


The FIBONACCI NUMBERS fn are
given by the recurrence :




fn
'
-
O
,
f, -
- I for =
fn it for -

z
(n 32 )
,
-




E -



g . ( first few terms) :
1,1 2,3 5,8 13,21
, , , ,
. . -




2.2 )
g (
E -
:




theoretical Phd
Every computer scientist
graduates one student every year , except the first year after

how 2020 ?
they graduate . If there was one theoretical computer scientist in 1975
, many are there in




Let tn be the number of theoretical computer scientists ( Tcs ) in year 1975 t n :




t.in?:: : : :at:: :. . . e)boun::r:n:i: ions
I
to =
given
-




E I new student fn
no Has recurrence
but different
= -


same as
, ,




.




tn C- 1975
En t t n 2020 n 4S
-
= -
-
-
.
,
n -
z


^ n


l l tag =

fab
existing TCS , New graduates




Eg ( 2.3 )
How there long 1
many ways are to walk from step 0 to step n of a staircase
taking or 2

steps at a time with no
backtracking .




f
fi fi
n =3
' ' '




ar
.




( L
(




o o
o




let
Wn
=
number of walks / steps to step n .




W I to 1
step

to walk
only
=
, one
way
at
Wz 2
to walk to step directly to stop step I
=

Two step 2 : it or
ways .




'



:




For
larger n we can either to
step and then to walk to
walk n 2
step directly step n or
-


,




step n
-
l and take one
step .




walk
Every arises this
way ,
and the
types are
disjoint .





(
^ ^



n -
I n -
l
Wn =
Wn .
. two -
z
but
boundary conditions so




2 z
Wn '
fret ,
n -
n -




Wn .
,
walks to
step n -
I Wn -
z
walks to
step n -
2

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