  Descent polynomials and peak polynomials, which enumerate permutations with
given descent and peak sets respectively, have recently received considerable
attention. We give several formulas for $q$-analogs of these polynomials which
refine the enumeration by the length of the permutations. In the case of
$q$-descent polynomials we prove that the coefficients in one basis are
strongly $q$-log concave, and conjecture this property in another basis. For
peaks, we prove that the $q$-peak polynomial is palindromic in $q$, resolving a
conjecture of Diaz-Lopez, Harris, and Insko.
