Exercises |
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 |
Exercises |