The five-vertex model and enumerations of plane partitions

We consider the five-vertex model on an $N\times2N$ lattice with fixed boundary conditions of a special type. We discuss a determinantal formula for the partition function in application to description of various enumrations of $N\times N\times (M-N)$ boxed plane partions. It is shown, that at the free-fermion point of the model, this formula reproduces MacMahon formula for the number of boxed plane partitions, while for generic weights (out of the free-fermion point) it describes enumerations with the weight depending on the cumulative number of jumps along vertical (or horisontal) rows. Various representations for the partition function, which describes such enumerations, are obtained.