Expansion in high dimension for the growth constants of lattice trees and lattice animals

We compute the first three terms of the 1/d expansions for the growth constants and one-point functions of nearest-neighbour lattice trees and lattice (bond) animals on the integer lattice Zd, with rigorous error estimates. The proof uses the lace expansion, together with a new expansion for the one-point functions based on inclusion-exclusion.Comment: 38 pages, 8 figures. Added section 6 to obtain the first term in the expansion, making the present paper more self-contained with very little change to the structure of the original paper. Accepted for publication in Combinatorics Probability and Computin

A polyominoes-permutations injection and tree-like convex polyominoes

AbstractPlane 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 an=6an−1−14an−2+16an−3−9an−4+2an−5, and compute the closed-form formula an=2n+2−(n3−n2+10n+4)/2