enumerative combinatorics

enumerative combinatorics

Enumerative combinatorics

deals with the question: if we know
that a set

S

is finite, how can we determine the exact number of

elements

that

S

contains?

Basic principles and techniques of enumerative combinatorics include:

•

the addition principle;

•

the

multiplication principle

;

•

the involution principle;

•

the

inclusion-exclusion principle

; and

•

the use of

generating functions

.

The principles listed above are disarmingly simple and seemingly
obvious. Nonetheless, when used properly they are powerful
tools for producing bijective proofs of combinatorial

identities

. On
the other hand, while generating functions can frequently be used to
give quick proofs of identities, it is sometimes difficult to extract
combinatorial proofs from such proofs.

The most fundamental principle of enumerative combinatorics and the
basis of all counting is the

addition principle

. It says that
if

S

is a

finite set

, then

|

S

|

=

∑

x

∈

S

1

.

More generally, if

S

=

A

⊔

B

, then

|

S

|

=

∑

x

∈

A

⊔

B

1

=

∑

x

∈

A

1

+

∑

x

∈

B

1

=

|

A

|

+

|

B

|

.

By

induction

on

n

, we also get

|

⊔

i

∈

[

n

]

S

i

|

=

∑

i

∈

[

n

]

|

S

i

|

.

As an application of the addition principle, we prove the

multiplication principle

.

Proposition 1

.

Multiplication principle.

If

S

and

T

are finite sets, then

|

S

×

T

|

=

|

S

|

⋅

|

T

|

.

Proof.

By the addition principle,

|

S

×

T

|

=

∑

(

x

,

y

)

∈

S

×

T

1

.

But we can write

S

×

T

as the

disjoint union

S

×

T

=

⊔

y

∈

T

S

×

{

y

}

.

The

projection map

S

×

{

y

}

→

S

is a

bijection

, so

|

S

×

{

y

}

|

=

|

S

|

. Hence it follows that

|

S

×

T

|

=

∑

y

∈

T

|

S

×

{

y

}

|

=

∑

y

∈

T

|

S

|

=

|

S

|

⋅

∑

y

∈

T

1

=

|

S

|

⋅

|

T

|

,

which

completes

the proof.
∎

The

involution principle

says that if

φ

:

S

→

S

is
an

involution

, then for any

X

⊂

S

,

|

X

|

=

|

φ

⁢

(

X

)

|

. The
following example illustrates the involution principle.

Proposition 2

.

Enumerating elements of the

powerset

.

Let

[

n

]

=

{

1

,

2

,

…

,

n

}

and let

2

[

n

]

be the powerset of

[

n

]

.
Then the

cardinality

of

2

[

n

]

is

2

n

.

Proof.

Define a

function

φ

:

2

[

n

]

→

2

[

n

]

by the

formula

φ

⁢

(

S

)

=

S

⁢

Δ

⁢

{

n

}

.

In other words,

φ

⁢

(

S

)

is the

symmetric difference

of

S

and

{

n

}

— if

n

is an element of

S

, we remove it, and if

n

is
not an element of

S

, we insert it. The function

φ

is an
involution, hence a bijection. Now notice that

2

[

n

-

1

]

⊂

2

[

n

]

and

φ

⁢

(

2

[

n

-

1

]

)

=

2

[

n

]

∖

2

[

n

-

1

]

. In
other words,

2

[

n

]

=

2

[

n

-

1

]

⊔

φ

⁢

(

2

[

n

-

1

]

)

.

Since

φ

is a bijection, this implies that

|

2

[

n

]

|

=

2

⋅

|

2

[

n

-

1

]

|

.

By induction on

n

we obtain

|

2

[

n

]

|

=

2

n

⋅

|

2

[

0

]

|

.

Now

[

0

]

=

∅

, so

2

[

0

]

=

{

∅

}

. Thus

|

2

[

0

]

|

=

1

, implying that

|

2

[

n

]

|

=

2

n

,

which is what we wanted to show.
∎

As we have seen above, it is possible to use the addition principle to
count the number of elements in the disjoint union of finite sets.
But what if we want to count the number of elements in a non-disjoint
union of finite sets? The

inclusion-exclusion principle

gives
us a way to do this. A common way of stating the inclusion-exclusion
principle is as the following

proposition

.

Proposition 3

.

Inclusion-exclusion principle.

Let

S

1

,

…

,

S

n

be finite sets. Then

|

⋃

i

∈

[

n

]

S

i

|

=

∑

T

∈

2

[

n

]

(

-

1

)

|

T

|

⁢

|

⋂

i

∈

T

S

i

|

.

The formula in this proposition uses negative numbers. But the core
of the inclusion-exclusion principle is a statement about

natural
numbers

.

Proposition 4

.

Let

S

and

T

be finite sets. Then

|

S

|

+

|

T

|

=

|

S

∪

T

|

+

|

S

∩

T

|

.

Thus the

lattice

of finite subsets of

N

is a

modular lattice

.

Proof.

By the addition principle,

|

S

|

+

|

T

|

=

|

(

S

∖

T

)

⊔

(

S

∩

T

)

|

+

|

(

T

∖

S

)

⊔

(

S

∩

T

)

|

=

|

S

∖

T

|

+

|

S

∩

T

|

+

|

T

∖

S

|

+

|

S

∩

T

|

.

Now observe that

S

∪

T

=

(

S

∖

T

)

⊔

(

S

∩

T

)

⊔

(

T

∖

S

)

. So by a second application of the addition principle,

|

S

∪

T

|

+

|

S

∩

T

|

=

|

S

∖

T

|

+

|

S

∩

T

|

+

|

T

∖

S

|

+

|

S

∩

T

|

.

Hence

|

S

|

+

|

T

|

=

|

S

∪

T

|

+

|

S

∩

T

|

.
∎

Title

enumerative combinatorics

Canonical name

EnumerativeCombinatorics

Date of creation

2013-03-22 16:45:58

Last modified on

2013-03-22 16:45:58

Owner

mps (409)

Last modified by

mps (409)

Numerical id

14

Author

mps (409)

Entry type

Topic

Classification

msc 05-00

Related topic

DijkstrasAlgorithm

Related topic

RuleOfProduct

Generated on Thu Feb 8 20:34:49 2018 by

LaTeXML