  We give a combinatorial description of a new diagram algebra, the partial
Temperley--Lieb algebra, arising as the generic centralizer algebra
$\mathrm{End}_{\mathbf{U}_q(\mathfrak{gl}_2)}(V^{\otimes k})$, where $V = V(0)
\oplus V(1)$ is the direct sum of the trivial and natural module for the
quantized enveloping algebra $\mathbf{U}_q(\mathfrak{gl}_2)$. It is a proper
subalgebra of the Motzkin algebra (the
$\mathbf{U}_q(\mathfrak{sl}_2)$-centralizer) of Benkart and Halverson. We prove
a version of Schur--Weyl duality for the new algebras, and describe their
generic representation theory.
