  By using permutation representations of maps, one obtains a bijection between
all maps whose underlying graph is isomorphic to a graph $G$ and products of
permutations of given cycle types. By using statistics on cycle distributions
in products of permutations, one can derive information on the set of all
$2$-cell embeddings of $G$. In this paper, we study multistars -- loopless
multigraphs in which there is a vertex incident with all the edges. The well
known genus distribution of the two-vertex multistar, also known as a dipole,
can be used to determine the expected genus of the dipole. We then use a result
of Stanley to show that, in general, the expected genus of every multistar with
$n$ nonleaf edges lies in an interval of length $2/(n+1)$ centered at the
expected genus of an $n$-edge dipole. As an application, we show that the face
distribution of the multistar is the same as the face distribution gained when
adding a new vertex to a $2$-cell embedded graph, and use this to obtain a
general upper bound for the expected number of faces in random embeddings of
graphs.
