Möbius inversion

Möbius inversion

Theorem 1

(Moebius Inversion).

Let

μ

be the

Moebius function

, and let

f

and

g

be two

functions

on the

positive integers

. Then the following two conditions are

equivalent

:

f

⁢

(

n

)

=

∑

d

|

n

g

⁢

(

d

)

⁢

for all

⁢

n

∈

ℕ

*

(1)

g

⁢

(

n

)

=

∑

d

|

n

μ

⁢

(

d

)

⁢

f

⁢

(

n

d

)

⁢

for all

⁢

n

∈

ℕ

*

(2)

Proof:

Fix some

n

∈

ℕ

*

. Assuming (1), we have

∑

d

|

n

μ

⁢

(

d

)

⁢

f

⁢

(

n

d

)

=

∑

d

|

n

μ

⁢

(

d

)

⁢

∑

e

|

n

/

d

g

⁢

(

e

)

=

∑

k

|

n

∑

d

|

k

μ

⁢

(

d

)

⁢

g

⁢

(

n

k

)

=

∑

k

|

n

g

⁢

(

n

k

)

⁢

∑

d

|

k

μ

⁢

(

d

)

=

g

⁢

(

n

)

,

the last step following from the

identity

given in the

Mobius function

entry.

Conversely

, assuming (2), we get

∑

d

|

n

g

⁢

(

d

)

=

∑

d

|

n

∑

e

|

d

μ

⁢

(

e

)

⁢

f

⁢

(

d

e

)

=

∑

k

|

n

f

⁢

(

n

k

)

⁢

∑

e

|

k

μ

⁢

(

k

e

)

=

f

⁢

(

n

)

(by the same identity)

as claimed.

Definitions:

In the notation above,

f

is called the Möbius transform
of

g

, and

formula

(2) is called the Möbius

inversion

formula.

Möbius-Rota inversion

G.-C. Rota has described a

generalization

of the Möbius

formalism

.
In it, the set

ℕ

*

, ordered by the

relation

x

|

y

between

elements

x

and

y

, is replaced by a more general ordered set, and

μ

is replaced by a function of two variables.

Let

(

S

,

≤

)

be a

locally finite

ordered set, i.e. an ordered set such that

{

z

∈

S

|

x

≤

z

≤

y

}

is a

finite set

for all

x

,

y

∈

S

. Let

A

be the set
of functions

α

:

S

×

S

→

ℤ

such that

α

⁢

(

x

,

x

)

=

1

⁢

for all

⁢

x

∈

S

(3)

α

⁢

(

x

,

y

)

≠

0

⁢

implies

⁢

x

≤

y

(4)

A

becomes a monoid if we define the

product

of any two of its
elements, say

α

and

β

, by

(

α

⁢

β

)

⁢

(

x

,

y

)

=

∑

t

∈

S

α

⁢

(

x

,

t

)

⁢

β

⁢

(

t

,

y

)

.

The sum makes sense because

α

⁢

(

x

,

t

)

⁢

β

⁢

(

t

,

y

)

is nonzero
for only finitely many values of

t

.
(Clearly this definition is akin to the definition of the product
of two square matrices.)

Consider the element

ι

of

A

defined simply by

ι

⁢

(

x

,

y

)

=

{

1

if

x

≤

y

0

otherwise.

The function

ι

, regarded as a matrix over

ℤ

, has an inverse
matrix, say

ν

. That means

∑

t

∈

S

ι

⁢

(

x

,

t

)

⁢

ν

⁢

(

t

,

y

)

=

{

1

if

x

=

y

,

0

otherwise.

Thus for any

f

,

g

∈

A

, the

equations

f

=

ι

⁢

g

(5)

g

=

ν

⁢

f

(6)

are equivalent.

Now let’s sketch out how the traditional Möbius inversion is a special
case of Rota’s notion. Let

S

be the set

ℕ

*

, ordered by the relation

x

|

y

between elements

x

and

y

. In this case,

ν

is essentially
a function of only one variable:

Proposition

3:

With the above notation,

ν

⁢

(

x

,

y

)

=

μ

⁢

(

y

/

x

)

for all

x

,

y

∈

ℕ

*

such that

x

|

y

.

The proof is fairly straightforward, by

induction

on the number
of elements of the interval

{

z

∈

S

|

x

≤

z

≤

y

}

.

Now let

g

be a function from

ℕ

*

to some

additive group

,
and write

g

¯

⁢

(

x

,

y

)

=

g

⁢

(

y

/

x

)

for all pairs

(

x

,

y

)

such
that

x

|

y

.

Example:

Let

E

be a set, and let

S

be the set of all
finite subsets of

E

, ordered by

inclusion

. The ordered set

S

is
left-finite, and for any

x

,

y

∈

S

such that

x

⊂

y

,
we have

ν

⁢

(

x

,

y

)

=

(

-

1

)

|

y

-

x

|

, where

|

z

|

denotes the

cardinal

of the finite set

z

.

A slightly more sophisticated example comes up in

connection

with the

chromatic polynomial

of a graph or

matroid

.

An Additional Generalization

A final generalization of

Moebius inversion

occurs when the sum is taken over all

integers

less than some real value

x

rather than over the

divisors

of an integer. Specifically, let

f

:

ℝ

→

ℂ

and define

F

:

ℝ

→

ℂ

by

F

⁢

(

x

)

=

∑

n

≤

x

f

⁢

(

x

n

)

.

Then

f

⁢

(

x

)

=

∑

n

≤

x

μ

⁢

(

n

)

⁢

F

⁢

(

x

n

)

.

Title

Möbius inversion

Canonical name

MobiusInversion

Date of creation

2013-03-22 11:46:58

Last modified on

2013-03-22 11:46:58

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

25

Author

mathcam (2727)

Entry type

Topic

Classification

msc 11A25

Synonym

Moebius inversion

Related topic

MoebiusFunction

Defines

Mobius inversion formula

Defines

Mobius transform

Defines

Mobius function

Defines

Mobius-Rota inversion

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