  The nonelliptic $\mathsf{A_2}$-webs with $k$ "$+$"s on the top boundary and
$3n-2k$ "$-$"s on the bottom boundary combinatorially model the space
$\mathsf{Hom}_{\mathfrak{sl}_3}(\mathsf{V}^{\otimes (3n-2k)},
\mathsf{V}^{\otimes k})$ of $\mathfrak{sl}_3$-module maps on tensor powers of
the natural $3$-dimensional $\mathfrak{sl}_3$-module $\mathsf{V}$, and they
have connections with the combinatorics of Springer varieties. Petersen,
Pylyavskyy, and Rhodes showed that the set of such $\mathsf{A_2}$-webs and the
set of semistandard tableaux of shape $(3^n)$ and type
$\{1^2,\dots,k^2,k+1,\dots, 3n-k\}$ have the same cardinalities. In this work,
we use the $\sf{m}$-diagrams introduced by Tymoczko and the Robinson-Schensted
correspondence to construct an explicit bijection, different from the one given
by Russell, between these two sets. In establishing our result, we show that
the pair of standard tableaux constructed using the notion of path depth is the
same as the pair constructed from applying the Robinson-Schensted
correspondence to a $3\,2\,1$-avoiding permutation. We also obtain a bijection
between such pairs of standard tableaux and Westbury's $\mathsf{A_2}$ flow
diagrams.
