Abstract :
We introduce new renements of the Bell, factorial, and unsigned Stirling numbers of
the rst 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.
