A polyominoes-permutations injection and tree-like convex polyominoes

Plane polyominoes are edge-connected sets of cells on the orthogonal lattice Z 2 , considered identical if their cell sets are equal up to an integral translation. We introduce a novel injection from the set of polyominoes with n cells to the set of permutations of [ n ] , and classify the families of convex polyominoes and tree-like convex polyominoes as classes of permutations that avoid some sets of forbidden patterns. By analyzing the structure of the respective permutations of the family of tree-like convex polyominoes, we are able to find the generating function of the sequence that enumerates this family, conclude that this sequence satisfies the linear recurrence a n = 6 a n − 1 − 14 a n − 2 + 16 a n − 3 − 9 a n − 4 + 2 a n − 5 , and compute the closed-form formula a n = 2 n + 2 − ( n 3 − n 2 + 10 n + 4 ) / 2 .