Some asymptotic results on multiple matching

We consider s randomly chosen permutations of the numbers 1,2,...,n, and write them under each other,thus forming an s x n matrix, called "random-batch". A rule, prescribing how many elements of a column may accur exactly one time, how many may occur exactly two times, etc., is called a fixpoint-structure. Assuming each possible permutation to be chosen with equal probability, the number of columns having a certain fixpoint-structure F is a random variable X(F). The limiting distribution of X(F) is considered for different cases of s=s(n). The main result (Theorem 4) says, that the common distribution of a finite number t of givenfixpoint-structures tends to the product of t Poisson-laws.