  We introduce a lifting of West's stack-sorting map $s$ to partition diagrams,
which are combinatorial objects indexing bases of partition algebras. Our
lifting $\mathscr{S}$ of $s$ is such that $\mathscr{S}$ behaves in the same way
as $s$ when restricted to diagram basis elements in the order-$n$ symmetric
group algebra as a diagram subalgebra of the partition algebra
$\mathscr{P}_{n}^{\xi}$. We then introduce a lifting of the notion of
$1$-stack-sortability, using our lifting of $s$. By direct analogy with Knuth's
famous result that a permutation is $1$-stack-sortable if and only if it avoids
the pattern $231$, we prove a related pattern-avoidance property for partition
diagrams, as opposed to permutations, according to what we refer to as
stretch-stack-sortability.
