  We present a formula for a generalisation of the Eulerian polynomial, namely
the joint distribution of major index and descent statistic on signed multiset
permutations. It allows a description in terms of the $h^*$-polynomial of a
certain polytope. We associate a family of polytopes to (generalised)
permutations of types $\mathsf{A}$ and $\mathsf{B}$. We use this connection to
study properties of the (generalised) Eulerian numbers, such as palindromicity
and unimodality, by identifying certain properties of the associated polytope.
We also present partial results on generalising the connection between descent
polynomials and polytopes to coloured (multiset) permutations.
