Exercises |

E:Check the equation for the cycle index ofS._{4}

E:Verify the details of locally finite partial orders.

E:Evaluate the cycle indicator polynomial of the action of the groupC,_{p}´C_{p}´C_{p}pbeing an odd prime, on itself by left multiplication. Evaluate also the cycle indicator polynomial of the action of the nonabelian group of orderpon itself by left multiplication.^{3}

E:Express the cycle indicatorC(Sin terms of the polynomials_{n},n)C(S._{k},k), 1 <= k <= n

E:Check that addition and convolution in fact define a ring structure on the setIof incidence functions._{F}(P)

E:Let(L,Ù,Ú)denote a lattice and(L, <= )the corresponding poset. We callfÎI_{F}(L)multiplicativeif and only if, for eachx,yinL, an order isomorphismimplies that[xÙy,xÚy] simeq [xÙy,x]´[xÙy,y]f(xÙy,xÚy)=f(xÙy,x)f(xÙy,y).

- Prove that the invertible multiplicative
fÎIform a group._{F}(L)- Show that the zeta function (and hence also the Moebius function) is multiplicative.
- Verify the description of the interval
[d,n]and the Moebius function on the lattice of divisors.

E:Prove the details in the examples.

E:Evaluate the characters of the natural actions ofSon_{X}Xand on^{n}X.^{n}_{i}

harald.fripertinger@kfunigraz.ac.at,

last changed: August 28, 2001

Exercises |