Möbius inversion
Theorem 1 (Moebius Inversion).
Let be the Moebius function, and let and be two functions on the positive integers. Then the following two conditions are equivalent:
(1) | |||||
(2) |
Proof: Fix some . Assuming (1), we have
the last step following from the identity given in the Mobius function entry.
Conversely, assuming (2), we get
as claimed.
Definitions: In the notation above, is called the Möbius transformof , 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 , ordered by the relation
between elements and , is replaced by a more general ordered set, and is replaced by a function of two variables.
Let be a locally finite ordered set, i.e. an ordered set such that is a finite set
for all . Let be the setof functions such that
(3) | |||||
(4) |
becomes a monoid if we define the product of any two of itselements, say and , by
The sum makes sense because is nonzerofor only finitely many values of .(Clearly this definition is akin to the definition of the productof two square matrices.)
Consider the element of defined simply by
The function , regarded as a matrix over , has an inversematrix, say . That means
Thus for any , the equations
(5) | |||||
(6) |
are equivalent.
Now let’s sketch out how the traditional Möbius inversion is a specialcase of Rota’s notion. Let be the set , ordered by the relation between elements and . In this case, is essentiallya function of only one variable:
Proposition 3: With the above notation, for all such that .
The proof is fairly straightforward, by induction on the numberof elements of the interval .
Now let be a function from to some additive group,and write for all pairs suchthat .
Example: Let be a set, and let be the set of allfinite subsets of , ordered by inclusion. The ordered set isleft-finite, and for any such that ,we have , where denotes the cardinalof the finite set .
A slightly more sophisticated example comes up inconnection 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 rather than over the divisors of an integer. Specifically, let and define by .
Then
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 |