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

TL;DR

This paper establishes a new upper bound on the per-tile entropy of n-ribbon tilings, showing it is at most log2 n, which improves previous bounds and enhances understanding of tiling complexity.

Contribution

The paper proves a tighter upper bound on the per-tile entropy of n-ribbon tilings, improving upon earlier bounds for general regions and rectangles.

Findings

Per-tile entropy is bounded above by log2 n.

Improved bounds over previous n-1 and log2(en) limits.

Provides theoretical insight into tiling enumeration complexity.

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.