Polyominoes defined by two vectors

The polyominoes' problem defined by two vectors has been proposed by M. Nivat in the course of the seminar held at the Dipartimento di Sistemi e Informatica di Firenze, on September 1992, on the subject Tiling the plane with a horizontal bar h m and a vertical bar v n . It is the problem of establishing the existence of a polyomino with a given number of cells in every column and every row. The problem is solved for the following classes of polyominoes: directed column-convex, directed convex, and parallelogram. The problem is also solved in the class of convex polyominoes in a particular case. Furthermore, it is possible to define an algorithm that controls the existence of a directed column-convex, or directed convex, or parallelogram polyomino, with a given number of cells in every column and in every row.