Reduced Latin rectangles and square matrices with identical sums in the rows and columns

A countable-parameter abstract generalization of the classical Riordan formula for 3-line Latin rectangles is given, where the suitable choice of parameters makes it possible to enumerate 3-line Latin rectangles with various features of the cyclical structure of their lines or equivalence classes.
As application there is presented the distributions of the values of the permanent on some classes of matrices.