  In their study of the equivariant K-theory of the generalized flag varieties
$G/P$, where $G$ is a complex semisimple Lie group, and $P$ is a parabolic
subgroup of $G$, Lenart and Postnikov introduced a combinatorial tool, called
the alcove paths model. It provides a model for the highest weight crystals
with dominant integral highest weights, generalizing the model by semistandard
Young tableaux. In this paper, we prove a simple and explicit formula
describing the crystal isomorphism between the alcove paths model and the
Gelfand-Tsetlin patterns model for type $A$.
