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单词 MobiusInversion
释义

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 equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath:

f(n)=d|ng(d) for all n*(1)
g(n)=d|nμ(d)f(nd) for all n*(2)

Proof: Fix some n*. Assuming (1), we have

d|nμ(d)f(nd)=d|nμ(d)e|n/dg(e)
=k|nd|kμ(d)g(nk)
=k|ng(nk)d|kμ(d)
=g(n),

the last step following from the identityPlanetmathPlanetmathPlanetmathPlanetmath given in the Mobius function entry.

Conversely, assuming (2), we get

d|ng(d)=d|ne|dμ(e)f(de)
=k|nf(nk)e|kμ(ke)
=f(n)  (by the same identity)

as claimed.

Definitions: In the notation above, f is called the Möbius transformof g, and formulaMathworldPlanetmathPlanetmath (2) is called the Möbius inversionPlanetmathPlanetmath formula.

Möbius-Rota inversion

G.-C. Rota has described a generalizationPlanetmathPlanetmath of the Möbius formalism.In it, the set *, ordered by the relationMathworldPlanetmathPlanetmathPlanetmath x|y between elementsx and y, is replaced by a more general ordered set, and μis replaced by a function of two variables.

Let (S,) be a locally finitePlanetmathPlanetmath ordered set, i.e. an ordered set such that{zS|xzy} is a finite setMathworldPlanetmath for all x,yS. Let A be the setof functions α:S×S such that

α(x,x)=1 for all xS(3)
α(x,y)0 implies xy(4)

A becomes a monoid if we define the productPlanetmathPlanetmathPlanetmath of any two of itselements, say α and β, by

(αβ)(x,y)=tSα(x,t)β(t,y).

The sum makes sense because α(x,t)β(t,y) is nonzerofor only finitely many values of t.(Clearly this definition is akin to the definition of the productof two square matrices.)

Consider the element ι of A defined simply by

ι(x,y)={1 if xy0otherwise.

The function ι, regarded as a matrix over , has an inversematrix, say ν. That means

tSι(x,t)ν(t,y)={1if x=y,0otherwise.

Thus for any f,gA, the equations

f=ιg(5)
g=νf(6)

are equivalent.

Now let’s sketch out how the traditional Möbius inversion is a specialcase of Rota’s notion. Let S be the set *, ordered by the relationx|y between elements x and y. In this case, ν is essentiallya function of only one variable:

PropositionPlanetmathPlanetmath 3: With the above notation, ν(x,y)=μ(y/x)for all x,y* such that x|y.

The proof is fairly straightforward, by inductionMathworldPlanetmath on the numberof elements of the interval {zS|xzy}.

Now let g be a function from * to some additive groupMathworldPlanetmath,and write g¯(x,y)=g(y/x) for all pairs (x,y) suchthat x|y.

Example: Let E be a set, and let S be the set of allfinite subsets of E, ordered by inclusion. The ordered set S isleft-finite, and for any x,yS such that xy,we have ν(x,y)=(-1)|y-x|, where |z| denotes the cardinalof the finite set z.

A slightly more sophisticated example comes up inconnectionMathworldPlanetmathPlanetmath with the chromatic polynomial of a graph or matroidMathworldPlanetmath.

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 divisorsMathworldPlanetmathPlanetmath of an integer. Specifically, let f: and define F: by F(x)=nxf(xn).

Then

f(x)=nxμ(n)F(xn).
TitleMöbius inversion
Canonical nameMobiusInversion
Date of creation2013-03-22 11:46:58
Last modified on2013-03-22 11:46:58
Ownermathcam (2727)
Last modified bymathcam (2727)
Numerical id25
Authormathcam (2727)
Entry typeTopic
Classificationmsc 11A25
SynonymMoebius inversion
Related topicMoebiusFunction
DefinesMobius inversion formula
DefinesMobius transform
DefinesMobius function
DefinesMobius-Rota inversion
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更新时间:2025/5/4 22:46:12