???????The component counts of random functions

Consider functions f : A -> A boolean OR C, where A and C are disjoint finite sets. The weakly connected components of the digraph of such a function are cycles of rooted trees, as in random mappings, and isolated rooted trees. Let n(1) = |A| and n(3) = |C|. When a function is chosen from all (n(1) +n(3))n(1) possibilities uniformly at random, then we find the following limiting behaviour as n(1) -> infinity. If n(3) = o(n(1)), then the size of the maximal mapping component goes to infinity almost surely; if n3 ~ gamma n(1), gamma > 0 a constant, then process counting numbers of mapping components of different sizes converges; if n(1) = o(n(3)), then the number of mapping components converges to 0 in probability. We get estimates on the size of the largest tree component which are of order log n3 when n(3) ~ gamma n1 and constant when n(3) ~ n(1)(alpha), alpha > 1. These results are similar to ones obtained previously for random injections, for which the weakly connected components are cycles and linear trees. (C) 2022 The Author(s). Published by Elsevier B.V.