Xiao Chuanqi Perfect matchings in polygonal chains

We will survey some results of the following type. A polyomino chain
is a planar square lattice that can be constructed by successively attaching
squares to the previous one in two possible ways. A random polyomino chain
is then generated by incorporating the Bernoulli distribution to the two types
of attachment, which describes a zeroth-order Markov process. Let ($\mathcal{R}_n, p$)
be the ensemble of random polyomino chains with n squares, where $p \in [0, 1]$
is a constant. We determine an explicit expression for the expectation of the
number of perfect matchings in $(\mathcal{R}_n, p)$.