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

Simon Blackburn; Yinsong Chen; Vladislav Kargin

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

Combinatorics 2024-12-09 v2

Authors: Simon Blackburn , Yinsong Chen , Vladislav Kargin

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.

Keywords

ribbon tilings per-tile entropy upper bound combinatorics statistical mechanics

Cite

@article{arxiv.2408.09272,
  title  = {An upper bound on the per-tile entropy of ribbon tilings},
  author = {Simon Blackburn and Yinsong Chen and Vladislav Kargin},
  journal= {arXiv preprint arXiv:2408.09272},
  year   = {2024}
}

Comments

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

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

Abstract

Keywords

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