Counting partial Latin rectangles and tridimensional rook placements with multisymmetric polynomials

We generalize Gessel's Formula for the number of Latin rectangles to partial Latin rectangles and non-attacking rook placements in a tridimensional chessboard. We also derive explicit short formulas for the generating series of the numbers of non-attacking rook placements on a chessboard with $2$ or $3$ levels. These series also count partial Latin rectangles with $2$ or $3$ rows.
The results are obtained following methods developed by MacMahon and Gessel for counting Latin squares and Latin rectangles, by means of scalar products of multisymmetric functions.