  This paper introduces a colored generalization of the Eulerian polynomials,
denoted the $\alpha$-colored Eulerian polynomials. We first compute these
polynomials by taking the $h$-vector of the $\alpha$-colored permutohedron, a
colored analog of the permutohedron which we develop. We also arrive at the
$\alpha$-colored Eulerian polynomials combinatorially by defining a new notion
of descent for colored permutations.
