  A quiver representation assigns a vector space to each vertex, and a linear
map to each arrow. When one considers the category
$\textrm{Vect}(\mathbb{F}_1)$ of vector spaces ``over $\mathbb{F}_1$'' (the
field with one element), one obtains $\mathbb{F}_1$-representations of a
quiver. In this paper, we study representations of a quiver over the field with
one element in connection to coefficient quivers. To be precise, we prove that
the category $\textrm{Rep}(Q,\mathbb{F}_1)$ is equivalent to the (suitably
defined) category of coefficient quivers over $Q$. This provides a conceptual
way to see Euler characteristics of a class of quiver Grassmannians as the
number of ``$\mathbb{F}_1$-rational points'' of quiver Grassmannians. We
generalize techniques originally developed for string and band modules to
compute the Euler characteristics of quiver Grassmannians associated to
$\mathbb{F}_1$-representations. These techniques apply to a large class of
$\mathbb{F}_1$-representations, which we call the
$\mathbb{F}_1$-representations with finite nice length: we prove sufficient
conditions for an $\mathbb{F}_1$-representation to have finite nice length, and
classify such representations for certain families of quivers. Finally, we
explore the Hall algebras associated to $\mathbb{F}_1$-representations of
quivers. We answer the question of how a change in orientation affects the Hall
algebra of nilpotent $\mathbb{F}_1$-representations of a quiver with bounded
representation type. We also discuss Hall algebras associated to
representations with finite nice length, and compute them for certain families
of quivers.
