A polyominoes-permutations injection and tree-like convex polyominoes

Plane polyominoes are edge-connected sets of cells on the orthogonal lattice Z2, 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=6a _n-1-14a _n-2+16a _n-3-9a _n-4+2a _n-5, and compute the closed-form formula a _n=2 ⁿ⁺²-(n ³-n ²+10n+4)/2.