Formulae for Polyominoes on Twisted Cylinders

λ > 4

“…Barequet et al proved the sequence (λ W ) is monotone increasing 5 and converges to λ. 3 The bigger W is thus the better (higher) the lower bound λ W on λ gets.…”

Polyominoes on twisted cylinders

“…1 It was also proven [1] that the sequence (λw) converges to λ. One can find in [1] formulae for the number of polyominoes on twisted cylinders obtained by building finite automata that model the "growth" of polyominoes on these cylinders, computing the transfer matrices of these automata, and deducing from these matrices the generating functions for the sequences that enumerate polyominoes on the twisted cylinders.…”

“…One can find in [1] formulae for the number of polyominoes on twisted cylinders obtained by building finite automata that model the "growth" of polyominoes on these cylinders, computing the transfer matrices of these automata, and deducing from these matrices the generating functions for the sequences that enumerate polyominoes on the twisted cylinders. This method proves that the formula for the number of polyominoes on a twisted cylinder of any width obeys a linear recurrence.…”

Two-stack-sorting with pop stacks

“…Figure 3: Small polyominoes in the plane Following [1] and [2], we consider polyominoes on a twisted cylinder of width w ∈ Z + . These polyominoes are drawn in the first quadrant of Z 2 by identifying all pairs of cells with coordinates (x, y) and (x − w, y + 1).…”