  Noticing that some recent variations of descent polynomials are special cases
of Carlitz and Scoville's four-variable polynomials, which enumerate
permutations by the parity of descent and ascent positions, we prove a
$q$-analogue of Carlitz-Scoville's generating function by counting the
inversion number and a type B analogue by enumerating the signed permutations
with respect to the parity of desecnt and ascent positions. As a by-product of
our formulas, we obtain a $q$-analogue of Chebikin's formula for alternating
descent polynomials, an alternative proof of Sun's gamma-positivity of her
bivariate Eulerian polynomials and a type B analogue, the latter refines
Petersen's gamma-positivity of the type B Eulerian polynomials.
