An upper bound on the per-tile entropy of ribbon tilings

Abstract

This paper considers $n$-ribbon tilings of general regions and their per-tile entropy (the binary logarithm of the number of tilings divided by the number of tiles). We show that the per-tile entropy is bounded above by ${\log }_{2}n$. This bound improves the best previously known bounds of $n-1$ for general regions, and the asymptotic upper bound of ${\log }_2(en)$ for growing rectangles, due to Chen and Kargin.

12 pages. Compared with the previous version: a typo corrected, a reference added, and another open problem stated