Combinatorial Theory

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.

Mathematics Subject Classifications: 05B45, 52C20

Keywords: Ribbon tilings, domino tilings, dimer tilings