Exercises |

E:Prove the following combinatorial principle: IfXandYare finite sets andRis a commutative ring, andj:Y´X -> R, thenå_{fÎYX}Õ_{xÎX}j(f(x),x)=Õ_{xÎX}å_{yÎY}j(y,x).

E:Derive Pólya's theorem directly, using the fact thatfÎYis fixed under^{X}gÎGif and only iffis constant on the cyclic factors ofbar (g).

E:Prove by induction thatå_{pÎSn }q^{l(p)}=[n]!.

E:Derive the formula from exercise by considering a transversal of the left cosets ofS. (Hint: Show that the permutations_{k}ÅS_{n\k}pinSwhich are increasing both on_{n}andkform such a transversal.)n\k

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 |