The partition function of the four-vertex model in inhomogeneous external field and trace statistics

The exactly solvable four-vertex model with the fixed boundary conditions in the presence of inhomogeneous linearly growing external field is considered. The partition function of the model is calculated and represented in the determinantal form. The established connection with the boxed plane partitions allows us to calculate the generating function of plane partitions with the fixed sums of their diagonals. The obtained results are another example of the connection of integrable models with the enumerative combinatorics.