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arXiv · 2305.06332

On enumeration and entropy of ribbon tilings

Yinsong Chen·Vladislav Kargin
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Abstract

The paper considers ribbon tilings of large regions and their per-tile entropy (the logarithm of the number of tilings divided by the number of tiles). For tilings of general regions by ribbon tiles of length $n$, we give an upper bound on the per-tile entropy as $n - 1$. For growing rectangular regions, we prove the existence of the asymptotic per-tile entropy and show that it is bounded from below by $\log_2 (n/e)$ and from above by $\log_2(en)$. For growing generalized "Aztec Diamond'' regions and for growing "stair'' regions, the asymptotic per-tile entropy is calculated exactly as $1/2$ and $\log_2(n + 1) - 1$, respectively.

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Yinsong Chen, Vladislav Kargin. 2023-05-10. On enumeration and entropy of ribbon tilings. https://doi.org/10.37236/10991

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