Enumerating Polyominoes with Fixed Perimeter Defect

Polycubes with Small Perimeter Defect

“…Our main result is that for each fixed non-negative value of k, the generating function of the sequence (A 3 (n, 4n + 2 − k)) is rational. This work generalizes a previous work [2], in which we investigated polyominoes (polycubes in two dimensions) with fixed defect.…”

Minimal-Perimeter Lattice Animals and the Constant-Isomer Conjecture

“…Asinowski et al [2] defined the excess of a perimeter cell as the number of adjacent occupied cell minus one, and the total perimeter excess of an animal Q, e P (Q), as the sum of excesses over all perimeter cells of Q. We extend this definition to border cells, and, in a similar manner, define the excess of a border cell as the number of adjacent empty cells minus one, and the border excess of Q, e B (Q), as the sum of excesses over all border cells of Q.…”

“…Proof: The sequence (n) is weakly-monotone increasing. 2 Assume that there exists a minimal-perimeter polyomino Q with a hole. Consider the polyomino Q that is obtained by filling this hole.…”

“…Asinowski et al [2,3] provided formulae for polyominoes and polycubes with perimeter size close to the maximum possible. On the other extreme reside animals with the minimum possible perimeter size for their area.…”

A Size‐Perimeter Discrete Growth Model for Percolation Clusters

“…As the enumeration of s n,m is still an open problem [26,29], I cannot be computed directly from (4). ere exist analytic formulae for both the minimum and maximum possible perimeter of a polyomino on the site square lattice as the function of its size [30][31][32]:…”