Counting tableaux with row and column bounds (Q1893990)

It is well known that the generating function for tableaux of a given skew shape with \(r\) rows where the parts in the \(i\)th 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 \(i\)th row are bounded by nondecreasing upper and lower bounds which depend on \(i\) and the parts in the \(j\)th 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 \((\alpha, \beta)\)-plane partitions.