E: Prove the following combinatorial principle: If X and Y are finite sets and R is a commutative ring, and j:Y´X -> R, thenåfÎYXÕxÎXj(f(x),x)=ÕxÎXåyÎY j(y,x).
E: Derive Pólya's theorem directly, using the fact that fÎYX is fixed under gÎG if and only if f is constant on the cyclic factors of bar (g).
E: Prove by induction thatåpÎSn ql(p)=[n]!.
E: Derive the formula from exercise by considering a transversal of the left cosets of SkÅSn\k. (Hint: Show that the permutations p in Sn which are increasing both on k and n\k form such a transversal.)
|harald.fripertinger "at" uni-graz.at||http://www-ang.kfunigraz.ac.at/~fripert/||UNI-Graz||Institut für Mathematik||UNI-Bayreuth||Lehrstuhl II für Mathematik|