Refinements of the Bell and Stirling numbers

‎‎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.

Keywords:

‎Bell numbers‎ , ‎Stirling numbers