Stirling numbers interpolation using permutations with forbidden subsequences

We present a family of number sequences which interpolates between the sequences Bn, of Bell numbers, and n!. It is defined in terms of permutations with forbidden patterns or subsequences. The introduction, as a parameter, of the number m of right-to-left minima yields an interpolation between Stirling numbers of the second kind S(n,m) and of the first kind (signless) c(n,m). Moreover, q-counting the restricted permutations by special inversions gives an interpolation between variants of the usual q-analogues of these numbers.