Counting tableaux with row and column bounds

It is well known that the generating function for tableaux of a given skew shape with r rows where the parts in the ith row are bounded by some nondecreasing upper and lower bounds which depend on i can be written in the form of a determinant of size r. We show that the generating function for tableaux of a given skew shape with r rows and c columns where the parts in the ith row are bounded by nondecreasing upper and lower bounds which depend on i and the parts in the jth column are bounded by nondecreasing upper and lower bounds which depend on j can also be given in determinantal form. The size of the determinant now is r + 2c. We also show that determinants can be obtained when the nondecreasingness is dropped. Subsequently, analogous results are derived for (α, β)-plane partitions.