Exercise Sheet 4 MT454 Combinatorics To be returned on 3rd November 2004

Exercise Sheet 4

MT454 Combinatorics

To be returned on 3rd November 2004

1. For each of the posets shown below:

(a) Write down the general form of a matrix corresponding to an element ofI(P).

(b) Verify directly (by multiplying matrices) thatI(P) is closed under multiplication.

(c) Calculate the M¨obius functionµof the poset by using the method just before Theorem 3.23. Check thatµ=i⁻¹ by computing the matrix product µi.

2. Letnbe a positive integer. Son=p^e₁¹p^e₂²· · ·p^e_r^r, wherep₁, p₂, . . . , p_r are distinct prime numbers and wheree₁, e₂, . . . , e_r are positive in-

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tegers. Prove that the M¨obius functionµofD(n) is defined by

µ(d, d⁰) =











1 ifd=d,⁰

(−1)^k ifddivides d⁰, andd⁰/dis a product of k distinct primes,

0 otherwise

for alld, d⁰ ∈ D(n). [Hint: Sheet 3 Question 4 might be useful!]

3. Letrandsbe functions from Nto R. Suppose that r(n) =X

d

s(d)

where the sum runs over the positive divisors ofn. Prove that s(n) =X

d

r(n/d)µ(d)

where again the sum runs over the positive divisors ofn, and where

µ(d) =







1 ifd= 1,

(−1)^k ifdis a product ofkdistinct primes,

0 otherwise.

[Hint: Use M¨obius inversion inD(n).]

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