Each connected component of a mapping $\{1,2,...,n\}\rightarrow\{1,2,...,n\}$ contains a unique cycle. The largest such component can be studied probabilistically via either a delay differential equation or an inverse Laplace transform. The longest such cycle likewise admits two approaches: we find an (apparently new) density formula for its length. Implications of a constraint -- that exactly one component exists -- are also examined. For instance, the mean length of the longest cycle is $(0.7824...)\sqrt n$ in general, but for the special case, it is $(0.7978...)\sqrt n$, a difference of less than $2\%$.