Fig. 1. (a) a column-convex (but not convex) polyomino; (b) a convex polyomino; for both, the bounding rectangle is shown dotted.

Fig. 2. (a) A directed-convex polyomino; (b) a parallelogram polyomino.

Fig. 3. The ECO operator on directed-convex polyominoes.

Fig. 4. The first levels of the generating tree of the operator .

Fig. 5. (a) The application of the operator ^′ to a Grand–Dyck path ending with a rise step; (b) the application of the operator ^′ to a Grand–Dyck path ending with a fall step.

Fig. 6. The first levels of the generating tree of the operator ^′.

Fig. 7. The matrix filled by the numbers a_n,k, n=0,…,5 and k=1,…,6, and the row sums.

Fig. 8. The operator _P for parallelogram polyominoes.

Fig. 9. A two-colored Grand–Motzkin path.

Fig. 10. Encoding of a two-colored Grand–Motzkin path with a binary matrix.

Fig. 11. The upper and lower paths for a directed convex polyomino.

Fig. 12. The two-colored Motzkin prefix associated with the polyomino in Fig. 11.

Fig. 13. The mapping of a prefix of H_p,q into a path of M_p,q.

Fig. 14. The (inverse) mapping of an M_p,q path into the path depicted in Fig. 12.

Fig. 15. A symmetric directed-convex polyomino and its coding.

Fig. 16. The path in H_p,q, and the Grand–Dyck path corresponding to the polyomino in Fig. 15.