We prove that the number of monomer-dimer tilings of an n×n square
grid, with m<n monomers in which no four tiles meet at any point is
m2m+(m+1)2m+1, when m and n have the same parity. In addition, we
present a new proof of the result that there are n2n−1 such tilings with
n monomers, which divides the tilings into n classes of size 2n−1. The
sum of these tilings over all monomer counts has the closed form
2n−1(3n−4)+2 and, curiously, this is equal to the sum of the squares of
all parts in all compositions of n. We also describe two algorithms and a
Gray code ordering for generating the n2n−1 tilings with n monomers,
which are both based on our new proof.Comment: Expanded conference proceedings: A. Erickson, M. Schurch, Enumerating
tatami mat arrangements of square grids, in: 22nd International Workshop on
Combinatorial Al- gorithms (IWOCA), volume 7056 of Lecture Notes in Computer
Science (LNCS), Springer Berlin / Heidelberg, 2011, p. 12 pages. More on
Tatami tilings at
http://alejandroerickson.com/joomla/tatami-blog/collected-resource