Discrete Mathematics

Discrete Mathematics

WEN-CHING LIEN Department of Mathematics National Cheng Kung University

2008

9.4: The exponential generating function

Recall:

For 0≤r ≤n,

C(n,r) = n!

r!(n−r)! =

1 r!



P(n,r)

where P(n,r)denotes the number of permutations of n objects taken r at a time.

So,

(1+x)ⁿ = C(n,0) +

n

X

k=1

C(n,k)x^k

= P(n,0) +

n

X

k=1

P(n,k)x^k k!

9.4: The exponential generating function

Recall:

For 0≤r ≤n,

C(n,r) = n!

r!(n−r)! =

1 r!



P(n,r)

where P(n,r)denotes the number of permutations of n objects taken r at a time.

So,

(1+x)ⁿ = C(n,0) +

n

X

k=1

C(n,k)x^k

= P(n,0) +

n

X

k=1

P(n,k)x^k k!

Definition

For a sequence a₀,a₁, · · · of real numbers,

f(x) =a₀+a₁x+a₂x²

2! +a₃x³

3! + · · · =

∞

X

i=0

a_ixⁱ i!

is called the exponential generating function for the given sequence.

Example:

e^x =1+x +x² 2! +x³

3! + · · ·

is the exponential generating function for the sequence 1,1,1, ...(and the ordinary generating function for the sequence 1,1,1/2!,1/3!..).

Definition

For a sequence a₀,a₁, · · · of real numbers,

f(x) =a₀+a₁x+a₂x²

2! +a₃x³

3! + · · · =

∞

X

i=0

a_ixⁱ i!

is called the exponential generating function for the given sequence.

Example:

e^x =1+x +x² 2! +x³

3! + · · ·

is the exponential generating function for the sequence 1,1,1, ...(and the ordinary generating function for the sequence 1,1,1/2!,1/3!..).

Note:

By the series expansions of e^x and e^−x, we can obtain

1. e^x +e^−x

2 = 1+ x²

2! +x⁴

4! + · · · , 2. e^x −e^−x

2 = x+x³

3! +x⁵

5! + · · · .

Example

A ship carries 48 flags, 12 each of the colors red, white, blue, and black. 12 of these flags are placed on a vertical pole in order to communicate a signal to other ships.

(a) How many of these signals use an even number of blue flags and an odd number of black flags?

(b) How many of the signals have at least 3 white flags or no white flags at all?

Example

A ship carries 48 flags, 12 each of the colors red, white, blue, and black. 12 of these flags are placed on a vertical pole in order to communicate a signal to other ships.

(a) How many of these signals use an even number of blue flags and an odd number of black flags?

(b) How many of the signals have at least 3 white flags or no white flags at all?

Example

A ship carries 48 flags, 12 each of the colors red, white, blue, and black. 12 of these flags are placed on a vertical pole in order to communicate a signal to other ships.

(a) How many of these signals use an even number of blue flags and an odd number of black flags?

(b) How many of the signals have at least 3 white flags or no white flags at all?

Thank you.