
\begin{table}
\begin{center}
\begin{tabular}{|c|c|l|}
\hline
\multicolumn{3}{|c|}{\textbf{Enumerating sequence: $\{2\}_{n\geq 2}$}}\\
\hline\hline

\emph{thanks to Proposition} &\emph{add the pattern} &\emph{to the patterns} \\
\hline

\ref{2} & $21-3$ & $\begin{array}{l}
  \{1-23, 2-13, 1-32, 3-12\}\ or \\
  \{1-23, 2-13, 1-32, 31-2\}\ or \\
  \{1-23, 2-13, 13-2, 3-12\}\ or \\
  \{1-23, 2-13, 1-32, 31-2\} \\
\end{array}$\\
\hline

\ref{2} & $21-3$ & $\begin{array}{l}
  \{1-23, 2-13, 2-31, 1-32\}\ or \\
  \{1-23 , 2-13 , 2-31 , 13-2\}\ or \\
  \{1-23 , 2-13 , 23-1 , 1-32\}\ or \\
  \{1-23 , 2-13 , 23-1 , 13-2\} \\
\end{array}$\\
\hline

\ref{2} & $21-3$ & $\begin{array}{l}
  \{1-23 , 2-13 , 2-31 , 3-12\}\ or \\
  \{1-23 , 2-13 , 2-31 , 31-2\}\ or \\
  \{1-23 , 2-13 , 23-1 , 3-12\}\ or \\
  \{1-23 , 2-13 , 23-1 , 31-2\} \\
\end{array}$\\
\hline

\ref{2} & $21-3$ & $\begin{array}{l}
  \{1-23 , 2-13 , 23-1 , 32-1\}\ or \\
  \{1-23 , 2-13 , 2-31 , 32-1\} \\
\end{array}$\\
\hline

\ref{2} & $21-3$ & $\begin{array}{l}
  \{2-13 , 2-31 , 1-32 , 3-12\}\ or \\
  \{2-13 , 2-31 , 1-32 , 31-2\}\ or \\
  \{2-13 , 2-31 , 13-2 , 31-2\} \\
\end{array}$\\
\hline

\ref{2} & $21-3$ & $\begin{array}{l}
  \{2-13 , 23-1 , 1-32 , 31-2\}\ or \\
  \{2-13 , 23-1 , 1-32 , 3-12\}\ or \\
  \{2-13 , 23-1 , 13-2 , 31-2\}\ or \\
  \{2-13 , 23-1 , 13-2 , 3-12\} \\
\end{array}$\\
\hline

\ref{3} & $3-12$ & $\begin{array}{l}
  \{1-23 , 2-13 , 2-31 , 31-2\}\ or \\
  \{1-23 , 2-13 , 23-1 , 31-2\} \\
\end{array}$\\
\hline

\ref{3} & $3-12$ & $\begin{array}{l}
  \{1-23 , 2-13 , 1-32 , 31-2\}\ or \\
  \{1-23 , 2-13 , 13-2 , 31-2\} \\
\end{array}$\\
\hline

\ref{3} & $3-12$ & $\begin{array}{l}
  \{1-23 , 2-13 , 2-31 , 31-2\}\ or \\
  \{1-23 , 2-13 , 23-1 , 31-2\} \\
\end{array}$\\
\hline

\ref{3} & $3-12$ & $\begin{array}{l}
  \{1-23 , 21-3 , 1-32 , 31-2\}\ or \\
  \{1-23 , 21-3 , 13-2 , 31-2\} \\
\end{array}$\\
\hline

\ref{3} & $3-12$ & $\begin{array}{l}
  \{1-23 , 2-31 , 1-32 , 31-2\}\ or \\
  \{1-23 , 2-31 , 13-2 , 31-2\} \\
\end{array}$\\
\hline

\ref{3} & $3-12$ & $\begin{array}{l}
  \{1-23 , 23-1 , 1-32 , 31-2\}\ or \\
  \{1-23 , 23-1 , 13-2 , 31-2\} \\
\end{array}$\\
\hline

\ref{4} & $23-1$ & $\begin{array}{l}
  \{2-13 , 2-31 , 1-32 , 31-2\} \\
\end{array}$\\
\hline

\ref{4} & $23-1$ & $\begin{array}{l}
  \{1-23 , 2-31 , 13-2 , 3-12\}\ or \\
  \{1-23 , 2-31 , 13-2 , 31-2\} \\
\end{array}$\\
\hline


\end{tabular}
\end{center}
\caption{permutations avoiding five patterns}\label{5mot_2(1)}
\end{table}

\begin{table}
\begin{center}
\begin{tabular}{|c|c|l|}
\hline
\multicolumn{3}{|c|}{\textbf{Enumerating sequence: $\{2\}_{n\geq 2}$}}\\
\hline\hline

\emph{thanks to Proposition} &\emph{add the pattern} &\emph{to the patterns} \\
\hline

\ref{4} & $23-1$ & $\begin{array}{l}
  \{1-23 , 2-31 , 1-32 , 3-12\}\ or \\
  \{1-23 , 2-31 , 1-32 , 31-2\} \\
\end{array}$\\
\hline

\ref{4} & $23-1$ & $\begin{array}{l}
  \{1-23 , 21-3 , 2-31 , 1-32\}\ or \\
  \{1-23 , 21-3 , 2-31 , 13-2\}\ or \\
  \{1-23 , 21-3 , 2-31 , 3-12\}\ or \\
  \{1-23 , 21-3 , 2-31 , 31-2\} \\
\end{array}$\\
\hline

\ref{4} & $23-1$ & $\begin{array}{l}
  \{1-23 , 2-13 , 2-31 , 1-32\}\ or \\
  \{1-23 , 2-13 , 2-31 , 13-2\}\ or \\
  \{1-23 , 2-13 , 2-31 , 3-12\}\ or \\
  \{1-23 , 2-13 , 2-31 , 31-2\} \\
\end{array}$\\
\hline

\ref{5} & $1-32$ & $\begin{array}{l}
  \{1-23 , 2-13 , 2-31 , 13-2\}\ or \\
  \{1-23 , 2-13 , 23-1 , 13-2\} \\
\end{array}$\\
\hline

\ref{5} & $1-32$ & $\begin{array}{l}
  \{1-23 , 2-13 , 13-2 , 3-12\}\ or \\
  \{1-23 , 2-13 , 13-2 , 31-2\} \\
\end{array}$\\
\hline

\ref{5} & $1-32$ & $\begin{array}{l}
  \{1-23 , 21-3 , 2-31 , 13-2\}\ or \\
  \{1-23 , 21-3 , 23-1 , 13-2\} \\
\end{array}$\\
\hline

\ref{5} & $1-32$ & $\begin{array}{l}
  \{1-23 , 21-3 , 13-2 , 3-12\}\ or \\
  \{1-23 , 21-3 , 13-2 , 31-2\} \\
\end{array}$\\
\hline

\ref{5} & $1-32$ & $\begin{array}{l}
  \{1-23 , 2-31 , 13-2 , 3-12\}\ or \\
  \{1-23 , 2-31 , 13-2 , 31-2\} \\
\end{array}$\\
\hline

\ref{5} & $1-32$ & $\begin{array}{l}
  \{1-23 , 23-1 , 13-2 , 3-12\}\ or \\
  \{1-23 , 23-1 , 13-2 , 31-2\} \\
\end{array}$\\
\hline

\ref{u1} & $12-3$ & $\begin{array}{l}
  \{1-23 , 2-13 , 2-31 , 1-32\}\ or \\
  \{1-23 , 2-13 , 2-31 , 13-2\}\ or \\
  \{1-23 , 2-13 , 23-1 , 1-32\}\ or \\
  \{1-23 , 2-13 , 23-1 , 13-2\} \\
\end{array}$\\
\hline

\ref{u1} & $12-3$ & $\begin{array}{l}
  \{1-23 , 2-13 , 2-31 , 3-12\}\ or \\
  \{1-23 , 2-13 , 2-31 , 31-2\}\ or \\
  \{1-23 , 2-13 , 23-1 , 3-12\}\ or \\
  \{1-23 , 2-13 , 23-1 , 31-2\} \\
\end{array}$\\
\hline

\ref{u1} & $12-3$ & $\begin{array}{l}
  \{1-23 , 2-13 , 1-32 , 3-12\}\ or \\
  \{1-23 , 2-13 , 1-32 , 31-2\} \\
\end{array}$\\
\hline

\ref{u2} & $12-3$ & $\begin{array}{l}
  \{1-23 , 21-3 , 23-1 , 3-12\}\ or \\
  \{1-23 , 21-3 , 23-1 , 31-2\} \\
\end{array}$\\
\hline

\ref{u2} & $12-3$ & $\begin{array}{l}
  \{1-23 , 21-3 , 2-31 , 1-32\}\ or \\
  \{1-23 , 21-3 , 23-1 , 1-32\} \\
\end{array}$\\
\hline

\ref{u2} & $12-3$ & $\begin{array}{l}
  \{1-23 , 21-3 , 2-31 , 3-12\}\ or \\
  \{1-23 , 21-3 , 2-31 , 31-2\} \\
\end{array}$\\
\hline

\end{tabular}
\end{center}
\caption{permutations avoiding five patterns}\label{5mot_2(2)}
\end{table}
\normalsize
