  Carlitz and Scoville introduced the polynomials $A_n(x,y|{\alpha},{\beta})$,
which we refer to as the $(\alpha, \beta)$-Eulerian polynomials. These
polynomials count permutations based on Eulerian-Stirling statistics, including
descents, ascents, left-to-right maxima, and right-to-left maxima. Carlitz and
Scoville obtained the generating function of $A_n(x,y|{\alpha},{\beta})$. In
this paper, we introduce a new family of polynomials,
$P_n(u_1,u_2,u_3,u_4|{\alpha},{\beta})$, defined on permutations, incorporating
descent-Stirling statistics including valleys, exterior peaks, right double
descents, left double ascents, left-to-right maxima, and right-to-left maxima.
By employing the grammatical calculus introduced by Chen, we establish the
connection between the generating function of
$P_n(u_1,u_2,u_3,u_4|{\alpha},{\beta})$ and the generating function of the
$(\alpha,\beta)$-Eulerian polynomials $A_n(x,y|{\alpha},{\beta})$ introduced by
Carlitz and Scoville. Using this connection, we derive the generating function
of $P_n(u_1,u_2,u_3,u_4|{\alpha},{\beta})$, which can be specialized to obtain
the $(\alpha,\beta)$-extensions of generating functions for peaks, left peaks,
double ascents, right double ascents and left-right double ascents given by
David-Barton, Elizalde and Noy, Entringer, Gessel, Kitaev and Zhuang. Moreover,
we establish two relations between $P_n(u_1,u_2,u_3,u_4|{\alpha},{\beta})$ and
$A_n(x,y|{\alpha},{\beta})$, which enable us to derive
$(\alpha,\beta)$-extensions of results of Stembridge, Petersen, Br\"and\'en,
and Zhuang. Specializing $(\alpha,\beta)$-extensions of Stembridge's formula
and the left peak version of Stembridge's formula allows us to derive the
$(\alpha,\beta)$-extensions of the tangent and secant numbers.
