  Partial permutohedra are lattice polytopes which were recently introduced and
studied by Heuer and Striker. For positive integers $m$ and $n$, the partial
permutohedron $\mathcal{P}(m,n)$ is the convex hull of all vectors in
$\{0,1,\ldots,n\}^m$ whose nonzero entries are distinct. We study the face
lattice, volume and Ehrhart polynomial of $\mathcal{P}(m,n)$, and our methods
and results include the following. For any $m$ and $n$, we obtain a bijection
between the nonempty faces of $\mathcal{P}(m,n)$ and certain chains of subsets
of $\{1,\dots,m\}$, thereby confirming a conjecture of Heuer and Striker, and
we then use this characterization of faces to obtain a closed expression for
the $h$-polynomial of $\mathcal{P}(m,n)$. For any $m$ and $n$ with $n\ge m-1$,
we use a pyramidal subdivision of $\mathcal{P}(m,n)$ to establish a recursive
formula for the normalized volume of $\mathcal{P}(m,n)$, from which we then
obtain closed expressions for this volume. We also use a sculpting process (in
which $\mathcal{P}(m,n)$ is reached by successively removing certain pieces
from a simplex or hypercube) to obtain closed expressions for the Ehrhart
polynomial of $\mathcal{P}(m,n)$ with arbitrary $m$ and fixed $n\le 3$, the
volume of $\mathcal{P}(m,4)$ with arbitrary $m$, and the Ehrhart polynomial of
$\mathcal{P}(m,n)$ with fixed $m\le4$ and arbitrary $n\ge m-1$.
