Mobius Inversion

Permutations that are in two of the sets AI, A2 , . . . , An -l contain two patterns. These patterns either share an element, such as the patterns 12 and 23, or have no element in common, such as the patterns 12 and 34. A permutation which contains the two patterns 12 and 34 can be regarded as a permutation of the n - 2 symbols {12, 34, 5, ... ,n}. Thus, IAI

n

A31

=

^{(n -} 2)!. A permutation that contains the two patterns 12 and 23 contains the pattern 123 and thus can be regarded as a permutation of the n - 2 symbols {123,4, ... ,n}. Hence, IAI n A21 = (n - 2)!. In general, we see that

I

Ai nAjl=(n-2)!

for each 2-subset {i, j} of {I, 2, ... , n - I}. More generally, we see that a permutation which contains k specified patterns from the list 12,23, ... , (n - l)n can be regarded as a permutation of n - k symbols, and thus that

for each k-subset {iI, i2, ... ,ik} of {I, 2, ... ,n - I}. Since, for each k = 1,2, ... ,n - 1, there are

(nk"l)

k-subsetss of {I, 2, ... ,n-1}, applying the inclusion-exclusion principle

we obtain the formula in the theorem. 0

Using the formula of Theorem 6.5.1, we calculate that

The numbers Ql, Q2, Q3,'" are closely related to the derangement numbers. Indeed, we have Qn = Dn

+

D n-^{l ,} (n ::::: 2). (See Exercise 23.) Thus, knowing the derangement numbers, we can calculate all the numbers Qn, (n ::::: 2). Since we have already seen in the preceding section that Ds

=

44, D6

=

265, we conclude that Q6

=

D6

+

Ds

=

265

+

44 = 309.

6.6 Mobius Inversion

This section includes more sophisticated mathematics than the other sections in this chapter.

The inclusion-exclusion principle is an instance of Mobius inversion on a finite⁴ partially ordered set. In order to set the stage for the generality of Mobius inversion, we first discuss a somewhat more general version of the inclusion-exclusion principle.

Let n be a positive integer and consider the set Xn

=

{I, 2, ... ,n} of n elements, and the partially ordered set (P(Xn), <;:;) o~ all subsets of Xn partially ordered by containment. Let

40ne can replace the property of being finite by a weaker property called locally finite, which asserts that, for all a and b with a :S b, the interval {x : a :S x :S b} is a finite set.

be a real-valued function defined on P(Xn ). We use F to define a new function

by

C(K) =

L

^{F(L), (K} ~ ^Xn ), (6.10)

L~K

where, as indicated, K is a subset of Xn and the summation extends over all subsets L of K. Mobius inversion allows one to invert equation (6.10) and to recover F from C; specifically, we have

F(K)

= L

(_l)IKI-ILIC(L), (K ~ Xn). (6.11)

L~K

Notice that F is obtained from C in (6.11) in a way similar to that in which C is obtained from F in (6.10); the only difference is that in (6.11) we insert in front of each term of the summation either a 1 or -1 depending on whether

IKI -ILl

is even or odd.

Let AI, A2, ... , An be subsets of a finite set S, and for a set K ~ {I, 2, ... , n}, define F(K) to be the number of elements of S that belong to exactly those sets Ai with i

rf.

K. Thus, for 8 E S, ⁸ is counted by F(K) if and only if

Then

for each i E K, and for each j

rf.

K.

C(K)

= L

^F(L)

L~K

counts the Humber of elements of S that belong to all of the sets Aj with j not in K and possibly other sets as well. Thus,

By (6.11),

F(K) =

L

(_l)IKI-ILIC(L). (6.12)

L~K

Taking K

=

{I, 2, ... ,n} in (6.12), we get

F(Xn)

= L

^{(_1)n-1LIC(L). (6.13)}

L~Xn

Now, F(Xn) counts the number of elements of S that belong only to those sets Ai with i

rf.

Xn; that is, F(Xn) is the number of elements of S that belong to none of the sets

6.6. MOBIUS INVERSION 185 AI, A2 , .•. , An and thus equals the number of elements contained in Al nA2n·· ·nAn.

Substituting into (6.13), we obtain

IAI n A2 n ... n Ani =

L

(_l)n-1LII nillL Ail,

L<;;Xn

or, equivalently, by replacing L with its complement in Xn and calling it J, IAI n A2 n .. · n Ani =

L

^{(-1)1111 niEJ Ail·}

J<;;Xn

(6.14) Equation (6.14) is equivalent to the formula for the inclusion-exclusion principle as given in Theorem 6.1.1.

We now replace (P(Xn ), <;;;;) with an arbitrary finite partially ordered set (X, ::;).

To derive the formula for Mobius inversion, we first consider functions of two variables.

Let F(X) be the collection of all real-valued functions f: X x X --t R,

with the property that f(x, y) =

°

^{whenever x}

^1.

y. Thus, f(x, y) can be different from

°

^{only when} x ::; y. We define the convolution product h = f

*

9 of two functions

f

and 9 in F(X) by

h( ) { L{z:x::;z::;y} f(x, z)g(z,y),

x,y

=

_0, ^{if x::; y,}

otherwise.

Thus, in the convolution product, to compute h(x, y) when x ::; y, we add up all products f(x, z)g(z, y) as z varies over all elements z between x and y in the given partial order. We leave it as an exercise to verify that the convolution product satisfies the associative law:

f*(g*h)=(j*g)*h, (j,g,hinF(X».

There are three special functions in F(X) of interest to us. The first is the Kro- necker delta function 6, given by

6(x,y) = { 1, 0,

if x = y otherwise.

Note that 6

*

f

=

f

*

6

=

f for all functions

f

^E F(X), and thus 6 acts as an identity function with respect to convolution product. The second is the zeta function <; defined by

<;(x,y) = { 1, 0,

if x::; y otherwise.

The zeta function is a representation of the poset (X,~) in that it contains all the information about which pairs x, y of elements satisfy x ~ y.

Let I be a function in :F(X) such that I(y, y)

#

0 for all y in X. We can inductively define a function 9 in :F(X) by first letting

1

g(y, y)

=

ICy, y)' (y EX), (6.15) and then letting

g(x, y) = - - ( - ) 1

L

g(x, z)/(z, y), (x

<

y). (6.16)

I

y, y {z:x9<y}

From (6.16), we get

L

g(x, z)/(z, y)

=

6(x, y), (x ~ y). ^(6.17)

{z:x~z~y}

Equation (6.17) tells us that

9

* 1=

6,

and therefore 9 is a left-inverse function of

I

with respect to the convolution product.

In a similar way, we can show that

I

has a right-inverse function h satisfying

Using the associative law for convolution product, we get

Thus, 9

=

hand 9 is an inverse function of

I.

In sum, every function

I

E :F(X) with I(y,y)

#

0 for all y in X has an inverse function g, inductively defined by (6.15) and (6.16), satisfying

9

* I = 1*

9

=

6.

The third special function we define is the Mobius function 1-£. Since ((y, y)

=

1 for all y EX, ( has an inverse, and we define 1-£ to be its inverse. Therefore,

1-£*( = 6, and so, applying (6.17) with

1=(

and 9 = J-t, we get

L

J-t(x,z)((z,y) = 6(x,y), (~~ y),

{z:x~z~y}

or, equivalently,

L

J-t(x, z) = 6(x, y), (x ~ y). ( 6.18)

{z:x~z~y}

6.6. MOBIUS INVERSION 187