On Random Partitions Induced by Random Maps

The partition lattice of the set [n] with respect to refinement is studied. Any map ϕ: [n] → [n], is associated with a partition of [n] by taking preimages of the elements. Assume that t partitions p₁, p₂,.. , p_t are chosen independently according to the uniform measure on the set of mappings [n] → [n]. It is shown that the probability for the coarsest refinement of all the partitions pi to be the finest partition {{1},.. , {n}} tends to 1 for any t ≥ 3 and to e^−1/2 for t = 2. It is also proved that the probability for the finest coarsening of the partitions pi to be the one-block partition tends to 1 as t(n) − log n→∞ and tends to 0 as t(n) − log n. The size of the maximal block of the finest coarsening of all the pi for a fixed t is also studied.