  In a recent paper (arXiv:2301.09744), Erickson and Hunziker consider
partitions in which the arm-leg difference is an arbitrary constant $m$. In
previous works, these partitions are called $(-m)$-asymmetric partitions.
Regarding these partitions and their conjugates as the highest weights, they
prove an identity yielding an infinite family of dimension equalities between
$\mathfrak{gl}_n$ and $\mathfrak{gl}_{n+m}$ modules. Their proof proceeds by
the manipulations of the hook content formula. We give a simple bijective proof
of their result.
