\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
\multicolumn{3}{|c|}{\textbf{Enumerating sequence: $\{0\}_{n\geq k}$}}\\
\hline\hline

\emph{name}&\emph{choose a pattern from the following to add} &\emph{to the symmetry class} \\
\hline \hline

O1& $12-3$&\\
O2& $1-32$&\\
O3& $13-2$&\\
O4& $3-12$&\\
O5& $31-2$&$\{1-23,2-13,3-21\}$ (C1)\\
O6& $21-3$&\\
O7& $2-31$&\\
O8& $23-1$&\\
O9& $32-1$&\\
\hline
O10& $12-3$&\\
O11& $1-32$&\\
O12& $13-2$&\\
O13& $3-12$&\\
O14& $31-2$&$\{1-23,23-1,32-1\}$ (C2)\\
O15& $2-13$&\\
O16& $21-3$&\\
O17& $2-31$&\\
O18& $3-21$&\\
\hline
O19& $12-3$&\\
O20& $13-2$&\\
O21& $3-12$&\\
O22& $31-2$&$\{1-23,2-13,32-1\}$ (C3)\\
O23& $21-3$&\\
O24& $2-31$&\\
\hline
O25& $31-2$&\\
O26& $1-32$&\\
O27& $23-1$&$\{1-23,12-3,3-21\}$ (C4)\\
O28& $32-1$&\\
\hline
O29& $1-32$&\\
O30& $13-2$&\\
O31& $3-12$&$\{1-23,21-3,3-21\}$ (C5)\\
O32& $31-2$&\\
O33& $23-1$&\\
\hline
O34& $13-2$&\\
O35& $3-12$&$\{1-23,21-3,32-1\}$ (C6)\\
O36& $2-31$&\\
\hline
O37& $13-2$&$\{1-23,2-31,32-1\}$ (C7)\\
\hline
\end{tabular}
\end{center}
\caption{permutations avoiding four patterns}\label{4mot_0}
\end{table}
