  The exact enumeration of pure dimer coverings on the square lattice was
obtained by Kasteleyn, Temperley and Fisher in 1961. In this paper, we consider
the monomer-dimer covering problem (allowing multiple monomers) which is an
outstanding unsolved problem in lattice statistics. We have developed the state
matrix recursion method that allows us to compute the number of monomer--dimer
coverings and to know the partition function with monomer and dimer activities.
This method proceeds with a recurrence relation of so-called state matrices of
large size. The enumeration problem of pure dimer coverings and dimer coverings
with single boundary monomer is revisited in partition function forms. We also
provide the number of dimer coverings with multiple vacant sites. The related
Hosoya index and the asymptotic behavior of its growth rate are considered.
Lastly, we apply this method to the enumeration study of domino tilings of
Aztec diamonds and more generalized regions, so-called Aztec octagons and
multi-deficient Aztec octagons.
