  In 1977 Carlitz and Scoville introduced the cycle $(\alpha,t)$-Eulerian
polynomials $A^{\mathrm{cyc}}_n(x,y, t\,|\,\alpha)$ by enumerating permutations
with respect to the number of excedances, drops, fixed points and cycles. In
this paper, we introduce a nine-variable generalization of the Eulerian
polynomials $A_n(u_1,u_2,u_3,u_4, f, g, t\,|\,\alpha, \beta)$ in terms of
descent based statistics of permutations and prove a connection formula between
these two kinds of generalized Eulerian polynomials. By exploring the
connection formula, we derive plainly the exponential generating function of
the latter polynomials and various $\gamma$-positive formulas for variants of
Eulerian polynomials. In particular, our results unify and strengthen the
recent results by Ji and Ji-Lin. In related work to the transition matrix
between the Specht and web bases, Hwang, Jang and Oh recently introduced the
web permutations, which can be characterised by cycle Andr\'e permutations. We
show that enumerating the latter permutations with respect to the number of
drops, fixed points and cycles gives rise to the normalised $\gamma$-vectors of
the $(\alpha,t)$-Eulerian polynomials. Our result generalizes and unifies
several known results in the literature.
