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Counting Polyominoes on Twisted Cylinders

Abstract : We improve the lower bounds on Klarner's constant, which describes the exponential growth rate of the number of polyominoes (connected subsets of grid squares) with a given number of squares. We achieve this by analyzing polyominoes on a different surface, a so-called $\textit{twisted cylinder}$ by the transfer matrix method. A bijective representation of the "states'' of partial solutions is crucial for allowing a compact representation of the successive iteration vectors for the transfer matrix method.
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Gill Barequet, Micha Moffie, Ares Ribó, Günter Rote. Counting Polyominoes on Twisted Cylinders. 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), 2005, Berlin, Germany. pp.369-374, ⟨10.46298/dmtcs.3446⟩. ⟨hal-01184435⟩

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