  Motivated by the way Japanese tatami mats are placed on the floor, we
consider domino tilings with a constraint and estimate the number of such
tilings of plane regions. We map the system onto a monomer-dimer model with a
novel local interaction on the dual lattice. We use a variant of the
Hamiltonian replica exchange Monte Carlo method and the multi-parameter
reweighting technique to study the model. The properties of the quantity are
studied beyond exact enumeration and combinatorial method. The logarithm of the
number of the tilings is linear in the boundary length of the region for all
the regions studied.
