Exercises |

E:is called_{G}Xk-fold transitiveif and only if the corresponding action ofGon the set ofXof injective mapping is transitive:^{ k}_{i}Prove that, in case of a transitive action| G \\X_{i}^{ k}| =1., this is equivalent to:_{G}X(k-1)_{Gx}(X \{x }) is-fold transitive.

E:Prove that| G |is divisible by[ | X | ]if_{k}is finite and_{G}Xk-fold transitive.

E:Show thatSis_{n}n-fold transitive onwhilenAis_{n}(n-2)-fold transitive on, but notn(n-1)-fold transitive, forn ³3.

E:Prove that| S._{ n}\\m^{n}_{s}| =[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 ofk-tupels(nsuch that_{1}, ...,n_{k})nand_{i}ÎNånis equal to_{i}=n[n+k-1 choose n].

E:Prove that the Stirling numbers of the second kind satisfy the equationwherex^{n}= å_{k=0}^{n}S(n,k)[x]_{k},[x]_{k}:=x(x-1) ·...·(x-k+1).

harald.fripertinger@kfunigraz.ac.at,

last changed: August 28, 2001

Exercises |