Formulae for polyominoes on twisted cylinders

Abstract

Polyominoes are edge-connected sets of cells on the square lattice ℤ2. We investigate polyominoes on a square lattice embedded on so-called twisted cylinders of a bounded width (perimeter) w. We prove that the limit growth rate of polyominoes of the latter type approaches that of polyominoes of the former type, as w tends to infinity. We also prove that for any fixed value of w, the formula enumerating polyominoes on a twisted cylinder of width w satisfies a linear recurrence whose complexity grows exponentially with w. By building the finite automaton that "grows" polyominoes on the twisted cylinder, we obtain the prefix of the sequence enumerating these polyominoes. Then, we recover the recurrence formula by using the Berlekamp-Massey algorithm. © 2014 Springer International Publishing.