A short combinatorial proof of dimension identities of Erickson and Hunziker

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 highest weights, they prove an identity yielding an infinite family of dimension equalities between ${\mathrm{gl}}_n$ and ${\mathrm{gl}}_{n+m}$ modules. Their proof proceeds by the manipulations of the hook content formula. We give a simple combinatorial proof of their result.