Exercises

### Exercises

E:   GX is called k- fold transitive if and only if the corresponding action of G on the set of X ki of injective mapping is transitive:
| G \\Xi k | =1.
Prove that, in case of a transitive action GX, this is equivalent to:
Gx(X \{x }) is (k-1)-fold transitive.
E: Prove that | G | is divisible by [ | X | ]k if GX is finite and k-fold transitive.
E: Show that Sn is n-fold transitive on n while An is (n-2)-fold transitive on n, but not (n-1)-fold transitive, for n ³3.
E: Prove that | S n \\mns | =[n-1 choose m-1].
E: Use the Principle of Inclusion and Exclusion in order to prove the identity for the recontre numbers.
E: Show that the number of k-tupels (n1, ...,nk) such that ni Î N and åni=n is equal to
[n+k-1 choose n].
E: Prove that the Stirling numbers of the second kind satisfy the equation
xn= åk=0nS(n,k)[x]k,
where [x]k:=x(x-1) ·...·(x-k+1).

 harald.fripertinger "at" uni-graz.at http://www-ang.kfunigraz.ac.at/~fripert/ UNI-Graz Institut für Mathematik UNI-Bayreuth Lehrstuhl II für Mathematik
last changed: January 19, 2005

 Exercises