On enumeration and entropy of ribbon tilings

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.

20 pages, 12 figures