  In this paper, we study Eulerian polynomials for permutations and signed
permutations of the multiset $\{1,1,2,2,\ldots,n,n\}$. Properties of these
polynomials, including recurrence relations and unimodality are discussed. In
particular, we give a unified proof of the fact that these polynomials are
unimodal with modes in the middle.
