关于带状平铺的计数与熵
On enumeration and entropy of ribbon tilings
AI总结：
本文研究带状平铺的每块熵，给出一般区域上界，证明矩形区域渐近熵存在并界于对数函数之间，精确计算两类特殊区域的熵。
摘要：
本文研究大区域的带状平铺及其每块平铺的熵（平铺数量的对数除以平铺块数）。对于长度为 $n$ 的带状平铺一般区域，我们给出了每块平铺熵的上界为 $n - 1$。对于增长的矩形区域，我们证明了渐近每块平铺熵的存在性，并表明其下界为 $\log_2 (n/e)$，上界为 $\log_2(en)$。对于增长的广义“Aztec Diamond”区域和增长的“阶梯”区域，渐近每块平铺熵分别精确计算为 $1/2$ 和 $\log_2(n + 1) - 1$。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
