On enumeration and entropy of ribbon tilings (Q6046222)

Summary: 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 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.