Entropy of full covering of the kagome lattice by straight trimers

Abstract

We consider the number of ways all the sites of a kagome lattice can be covered by non-overlapping linear rigid rods where each rod covers 3 sites. We establish a 2-to-1 correspondence between the configurations of trimers on the kagome lattice to the covering by dimers of a related hexagonal lattice to show that entropy of coverings per trimer $s_{\text{tri,kag}}$ equals the entropy per dimer $ s_{\text{dim,hex}} $, and is given by $ s_{\text{tri,kag}} = s_{\text{dim,hex}} = \frac{1}{2 \pi} \int_0^{ 2 \pi/3} \log( 2 + 2 \cos k) dk \approx 0.323065947\ldots$.