  We introduce new refinements of the Bell, factorial, and unsigned Stirling
numbers of the first and second kind that unite the derangement, involution,
associated factorial, associated Bell, incomplete Stirling, restricted
factorial, restricted Bell, and $r$-derangement numbers (and probably more!).
By combining methods from analytic combinatorics, umbral calculus, and
probability theory, we derive several recurrence relations and closed form
expressions for these numbers. By specializing our results to the classical
case, we recover explicit formulae for the Bell and Stirling numbers as sums
over compositions.
