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We investigate quiver representations over $\mathbb{F}_1$. Coefficientquivers are combinatorial gadgets equivalent to $\mathbb{F}_1$-representationsof quivers. We focus on the case when the quiver $Q$ is a pseudotree. For suchquivers, we first use the notion of coefficient quivers to provide a completeclassification of asymptotic behaviors of indecomposable representations over$\mathbb{F}_1$. Then, we prove some fundamental structural results about theLie algebras associated to pseudotrees. Finally, we construct examples of$\mathbb{F}_1$-representations $M$ of a quiver $Q$ by using coverings, underwhich the Euler characteristics of the quiver Grassmannians$\textrm{Gr}^Q_{\underline{d}}(M)$ can be computed in a purely combinatorialway.

On the Combinatorics of $\mathbb{F}_1$-Representations of Pseudotree Quivers