Exercises

Exercises

E:   _GX is called k- fold transitive if and only if the corresponding action of G on the set of X ^k_i of injective mapping is transitive:

| G \\X_i ^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 S_n is n-fold transitive on n while A_n is (n-2)-fold transitive on n, but not (n-1)-fold transitive, for n ³3.

E: Prove that | S _n \\mⁿ_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 of k-tupels (n₁, ...,n_k) such that n_i Î N and ån_i=n is equal to

[n+k-1 choose n].

E: Prove that the Stirling numbers of the second kind satisfy the equation

xⁿ= å_k=0ⁿS(n,k)[x]_k,

where [x]_k:=x(x-1) ·...·(x-k+1).

Exercises