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 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.
@article{10_37236_10991,
author = {Yinsong Chen and Vladislav Kargin},
title = {On enumeration and entropy of ribbon tilings},
journal = {The electronic journal of combinatorics},
year = {2023},
volume = {30},
number = {2},
doi = {10.37236/10991},
zbl = {1518.52012},
url = {http://geodesic.mathdoc.fr/articles/10.37236/10991/}
}
TY - JOUR
AU - Yinsong Chen
AU - Vladislav Kargin
TI - On enumeration and entropy of ribbon tilings
JO - The electronic journal of combinatorics
PY - 2023
VL - 30
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/10991/
DO - 10.37236/10991
ID - 10_37236_10991
ER -
%0 Journal Article
%A Yinsong Chen
%A Vladislav Kargin
%T On enumeration and entropy of ribbon tilings
%J The electronic journal of combinatorics
%D 2023
%V 30
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/10991/
%R 10.37236/10991
%F 10_37236_10991
Yinsong Chen; Vladislav Kargin. On enumeration and entropy of ribbon tilings. The electronic journal of combinatorics, Tome 30 (2023) no. 2. doi: 10.37236/10991
