Exercises

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, then

å_fÎY^XÕ_xÎXj(f(x),x)=Õ_xÎXå_yÎY j(y,x).

E: Derive Pólya's theorem directly, using the fact that fÎY^X is fixed under gÎG if and only if f is constant on the cyclic factors of bar (g).

E: Prove by induction that

å_pÎSn q^l(p)=[n]!.

E: Derive the formula from exercise by considering a transversal of the left cosets of S_kÅS_n\k. (Hint: Show that the permutations p in S_n which are increasing both on k and n\k form such a transversal.)

Exercises