Strata of Random Mappings – a Combinatorial Approach

Let Fn denote the set of all mappings φ : {1, . . . , n} → {1, . . . , n} and assume that this set is equipped with the uniform distribution. Then a mapping φ ∈ Fn is usually called a random mapping. For our investigations it is convenient to represent random mappings by its functional graph Gφ, i.e. the graph consisting of the nodes 1, 2, . . . , n and of the edges (i, φ(i)), i = 1, . . . , n. It is easy to see that each component of such a graph consists of exactly one cycle of length ≥ 1 each point of which is the root of a labeled tree. Thus for each point x ∈ Gφ there exists a unique path connecting x with the next cyclic point. The length of this path is called the distance of x to the cycle. The set of all points at a fixed distance r from the cycle is often called the r-th stratum of φ. Let Ln(r) denote the number of nodes in the r-th stratum of a random mapping φ ∈ Fn. The behavior of this random variable for n → ∞ has attracted the interest of many authors. Harris [14] showed that the number of cyclic points Ln(0)/ √ n weakly converges to a Rayleigh distribution with mean value √