  We establish a connection between planar rook algebras and tensor
representations $\VV^{\otimes k}$ of the natural two-dimensional representation
$\VV$ of the general linear Lie superalgebra $\gl$. In particular, we show that
the centralizer algebra $\maths{End}_{\gl}(\VV^{\otimes k})$ is the planar rook
algebra $\CC \mathsf{P}_{k-1}$ for all $k \geq 1$, and we exhibit an explicit
decomposition of $\VV^{\otimes k}$ into irreducible $\gl$-modules. We obtain
similar results for the quantum enveloping algebra $\UU_\qq(\gl)$ and its
natural two-dimensional module $\VV_\qq$.
