Exponential Generating Functions

1

COS 341 Discrete Mathematics

Exponential Generating

Functions

Generating Functions

2 0 0 1

( , , , ) : sequence of real numbers

of this sequence is

the power serie

Gene

s

rating Function

( )

i i i

a

a

a

x

x

a

a

∞ =

=

∑

⋅

…

Ordinary

Ordinary

∧

3

Exponential Generating Functions

2

0 0 1

Exponential Generating func

( , , , ) : sequence of real numbers

of this

sequence is the power se

( )

ries

!

tion

i i i

a a

a

a x

x

i

a

∞ =

=

∑

⋅

…

Exponential generating function

examples

What is the generating function for the sequence (1,1,1,1,…)?

2 3 0 1 1 ! 1! 2! 3! i x i x x x x e i = ∞ = + + + + =

∑

What is the generating function for the sequence (1, 2, 4,8,…)?

2 3 2 0 2 2 (2 ) (2 ) 1 ! 1! 2! 3! i i x i x x x x e i = ∞ = + + + + =

∑

5

Operations on exponential generating

functions

• Addition

• Multiplication by fixed real number

• Shifting the sequence to the right

• Shifting to the left

0 0 1 1 has generating function

(a +b a, +b, )… a x( )+b x( )

0 1 has generating functi

(αa ,αa, )… on αa x( ) 0 1 has generating fu (0, 0, , , ) nction n ( ) n a a x a x × … … 1 0 1 has generating f ( unction ) ( , , ) k _i i i k k n a x a x a a x − = + −

∑

⋅ …

• Substituting

α

x for x

• Substitute x

n

_{for x}

0 1 2 1 1 has ge ( , 0, 0, , 0, 0, ) nerating function ( )n n n a a a a x −× −× … … … 2

0 1 2 has generating funct

( ,a α αa, a …) iona x( )α

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• Differentiation

• Integration

• Multiplication of generating functions

1 2 3 has generating function

( , 2 ,3a a a ) d a x( ) (ora x'( )) dx … 1 1 0 2 1 3 2 0

has generating function

(0, , , ) ( ) x a a a …

∫

f t dt

(

0

)(

0

) (

0

)

0 n n n n n n n n n n n k k n k a x b x c x c a b ∞ ∞ ∞ = = = − = ⋅ ⋅ = ⋅ = ⋅

∑

∑

∑

∑

Differentiation

0 1 2

is the exponential generating function f

( ) or ( , , , )

a x a a a …

Differentiation is equivalent to shifting the sequence to the left

0 ( ) ! i i i a a x x i = ∞ =

∑

⋅ 1 1 0 1 ( ) ! ( 1)! i i i i i i a a d a x i x x dx i i − − = = ∞ ∞ = ⋅ ⋅ = ⋅ −

∑

∑

1 2 3

is the exponential generating function for

( ) ( , , , )

d _{a x a a a}

9

Integration

0 1 2

is the exponential generating function f

( ) or ( , , , )

a x a a a …

Integration is equivalent to shifting the sequence to the right

0 ( ) ! i i i a a x x i = ∞ =

∑

⋅ 0 1 2 0

is the exponential generating function

( ) for (0, , , , ) x a t dt a a a

∫

… 0 ( ) x a t dt

∫

1 0 ! ( 1) i i i a x i i + = ∞ = ⋅ +

∑

0 ! ₀ x i i i a t dt i = ∞ =

∑ ∫

⋅ 1 0 ( 1)! i i i a x i + = ∞ = ⋅ +

∑

Multiplication

(

)(

) (

)

0 0 0 0 n n n n n n n n n n n k k n k c a b a x b x c x − ∞ ∞ ∞ = = = = ⋅ ⋅ = ⋅ = ⋅

∑

∑

∑

∑

0 0 0 0 ! ! ! ! ! ( )! n n n n n n n n k n n n k n k c a b n k n k a b c x x x n n n ∞ ∞ ∞ = − = = =  _  _ _  _{⋅ } _{⋅ =} _{⋅ }  _ _  _         = ⋅ −

∑

∑

∑

∑

0 ! !( )! n k n k k n n a b n c k k − = = ⋅ −

∑

0 n k n k k n a b k − =    = _{  } ⋅  

∑

Ordinary

Exponential

11

Implications of product rule

0 n n _k k n k c =

∑

₌ a b⋅ − 0 n n k n k k n c a b k − =    = _{  } ⋅  

∑

Ordinary

Exponential

( )

( ) ( )

C x

=

A x B x

Useful for counting with indistinguishable objects

Useful for counting with ordered objects

Interpretation of Multiplication: Product Rule

: number of arrangements of type A for objects : number of arrangements of type B for objects

n n

a n

b n

Given arrangements of type and type , define arrangements of type C for labeled objects as follows: Divide the group of labeled objects into two groups, the group and the

B Secon gr A First oup; r d ar n n

ange the group by an arrangement of type and the Sec group by an arrangementof

First

typ

ond

A

13

: number of arrangements of type A for people : number of arrangements of type B for people : number of arrangements of type C

for people n n n a n b n c n 0 n n k n k k n c a b k − =    = _{  } ⋅  

∑

( ) ( ) ( ) C x =A x B x

: exponential generating function : exponential generating function : exponential gener ( ) ( ) ating function ( ) n n n a A x b B x c C x

Interpretation of Multiplication: Product Rule

0

: exponential generating function for arra

( ) 0

ngements of type A : no empty group allowed

A x a =

Define arrangements of type D for labeled objects as follows: Divide the group of labeled objects into groups,

the group, group, , group arrange each group by an arra

First Second th ( 0,1,2, ) n k k n n k= … … gement of type A .

: exponential generating function for arrangements of ty D

( ) pe

D x

15

: exponential generating function for arrangements of type D with exactly groups

( ) k D x k ( ) ( )k k D x =A x 0 ( ) k( ) k D x D x = ∞ =

∑

0 ( )k k A x = ∞ =

∑

1 1 A x( ) = − 0

: exponential generating function for arra

( ) 0

ngements of type A : no empty group allowed

A x a =

Define arrangements of type for labeled objects as follows: Divide the group of labeled objects into groups,

and arrange each group by an arrangement of type (the groups are not numbere ) .

A E d k n n

: exponential generating function for arrangements of ty

( E) pe

17

: exponential generating function for arrangements of type with exactly ( groups ) E k E x k ( ) ( ) ! k k A x E x k = 0 ( ) k( ) k E x E x = ∞ =

∑

0 ( ) ! k k A x k = ∞ =

∑

( ) A x

e

=

Example

How many sequences of letters can be formed from A, B, and C such that the number of A's is odd and the number of B's is odd ?

n

odd

EGF for A's

! n n x n =

∑

2 x x

e

₋

e

− = EGF for B's 2 x x

e

₋

e

− = 0 EGF for C' ! s n n x x n

e

= ∞ = =

∑

2 required EG 2 F _

e

x−

e

−x__

e

x   = 3 4

2

x x x

e

₋

e

₊

e

− = 19

Example

How many sequences of letters can be formed from A, B, and C such that the number of A's is odd and the number of B's is odd ? n 3 required EGF 4

2

x x x

e

e

e

− = − + coefficient of 1 3 ( 1) ! 4

2

n n n x n  _{− + −} _  _  _     = required number 3 ( 1) 4

2

n_{− + −} n =

Example

How many ways can people be arranged into pairs, (the pairs are not numbered) ?

n

: exponential generating function for a single r

( ) pai A x 2 1, 0i fori 2 a = a = ≠ 2 ( ) 2 x A x =

: exponential generating function for arranging people into ( s ir ) pa E x n 2 2 ( ) x E x =

_e

/ 2 2 n ₁ n ₌e n₌  x _{e =} n!

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Derangements (or Hatcheck lady revisited)

: number of permutations on objects without a fixed poi t n n

d n

exponential generating function for number of derang ( ) : emen st

D x

A permutation on can be constructed by picking a subset of constructing a derangement of K and fixing the elements of .

Every permutation of [n] arises exact

[ ] [ ] [ ly on , ce t ]-his way. n K n n K 0

EGF for all permutatio ! 1

1 s n ! n n n x n x = ∞ = = −

∑

0

EGF for permutations with all elem 1

! ents fixed n x n x n

e

= ∞ =

∑

= 1 _{( )} 1 x D x x= ⋅

e

−

Derangements

1 _{( )} 1 x D x x= ⋅

e

− 1 ( ) 1 x D x x

e

− = − 0 0 ( 1) ! n n n n n x x n ∞ = = ∞  _ _  _ =_ − __ _   

∑



∑

0 ( coefficient o ) ! f ! 1 n k n k n d n x = k  ₋ _  _ = =_{ } 

∑

 0 ( 1) ! ! k n k n d n k =  ₋ _  _ = _{ } 

∑

