  In the present article we introduce two new combinatorial interpretations of
the $r$-Whitney numbers of the second kind obtained from the combinatorics of
the differential operators associated to the grammar $G:=\{ y\rightarrow
yx^{m}, x\rightarrow x\}$. By specializing $m=1$ we obtain also a new
combinatorial interpretation of the $r$-Stirling numbers of the second kind.
Again, by specializing to the case $r=0$ we introduce a new generalization of
the Stirling number of the second kind and through them a binomial type family
of polynomials that generalizes Touchard's. Moreover, we show several
well-known identities involving the $r$-Dowling polynomials and the $r$-Whitney
numbers using the combinatorial differential calculus. Finally we prove that
the $r$-Dowling polynomials are a Sheffer family relative to the generalized
Touchard binomial family, study their umbral inverses, and introduce
$[m]$-Stirling numbers of the first kind. From the relation between umbral
calculus and the Riordan matrices we give several new combinatorial identities
involving the $r$-Whitney number of both kinds, Bernoulli and Euler
polynomials.
