  Lomonaco and Kauffman introduced knot mosaic system to give a definition of
quantum knot system. This definition is intended to represent an actual
physical quantum system. A knot $(m,n)$-mosaic is an $m \times n$ matrix of
mosaic tiles which are $T_0$ through $T_{10}$ depicted as below, representing a
knot or a link by adjoining properly that is called suitably connected. An
interesting question in studying mosaic theory is how many knot $(m,n)$-mosaics
are there. $D_{m,n}$ denotes the total number of all knot $(m,n)$-mosaics. This
counting is very important because the total number of knot mosaics is indeed
the dimension of the Hilbert space of these quantum knot mosaics.
  In this paper, we find a table of the precise values of $D_{m,n}$ for $4 \leq
m \leq n \leq 6$ as below. Mainly we use a partition matrix argument which
turns out to be remarkably efficient to count small knot mosaics.
  \begin{center} \begin{tabular}{|c|r|r|r|} \hline $D_{m,n}$ & $n=4$ & $n=5$ &
$n=6$ \\ \hline $m=4$ & $2594$ & $54,226$ & $1,144,526$ \\ \hline $m=5$ & &
$4,183,954$ & $331,745,962$ \\ \hline $m=6$ & & & $101,393,411,126$ \\ \hline
\end{tabular} \end{center}
