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\parskip=12pt

\begin{document}

\section*{Appendix}

In the following $C^7$ denotes a chain of rank $7$

\subsection*{A.1 $P\left(6,\left\{[1,2],[2,6] \right\}\right)+P\left(6,\left\{[2,5],[5,6] \right\}\right)$}

Take $\HD^{N}_{[1,2]}\HD^{N}_{[2,6]}\left(C^7\right)$ and 
$\HD^{N}_{[1,5]}\HD^{N}_{[5,6]}\left(C^7\right)$. Identify the elements of both
posets at rank $1$ and at rank $6$. Figure \ref{F_lpos1} represents the
resulting poset for $N=2$. 


\begin{figure}[h]
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\caption{$P\left(6,\left\{[1,2],[2,6] \right\}\right)+P\left(6,\left\{[2,5],[5,6] \right\}\right)$}
\label{F_lpos1}
\end{figure}

\newpage

\subsection*{A.2 $P\left(6,\left\{[1,3],[3,4],[4,6]\right\}\right)
+P\left(6,\left\{[1,2],[2,3]\right\}\right)
+P\left(6,\left\{[4,5],[5,6] \right\}\right)$}

Take
\begin{eqnarray*}
P^I(N)&=&
\HD^{N}_{[1,3]}\HD^{N}_{[3,4]}\HD^{N}_{[4,6]}\HD^{N+1}_{[4,5]}
\left(C^7\right)\\ 
P^{II}(N)&=&
\HD^{N+1}_{[1,2]}\HD^{N}_{[1,6]}\HD^{N}_{[2,4]}\left(C^7\right),
\quad\mbox{and}\\   
P^{III}(N)&=&
\HD^{N}_{[1,5]}\HD^{N}_{[3,5]}\HD^{N}_{[5,6]}\left(C^7\right).\\
\end{eqnarray*}
Identify the elements of $P^{I}(N)$ with the elements of $P^{II}(N)$ at
ranks $1,4,5$, and $6$. 
Identify the elements of $P^{I}(N)$ with the elements of $P^{III}(N)$ at
ranks $1,2,3$, and $6$. Figure \ref{F_lpos2} represents the resulting poset for
$N=2$. 




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\caption{$P\left(6,\left\{[1,3],[3,4],[4,6]\right\}\right)
+P\left(6,\left\{[1,2],[2,3]\right\}\right)
+P\left(6,\left\{[4,5],[5,6] \right\}\right)$}
\label{F_lpos2}
\end{figure}

\newpage

\subsection*{A.3 $P\left(6,\left\{[1,2],[3,4],[4,5]\right\}\right)
+P\left(6,\left\{[3,5],[5,6]\right\}\right)
+P\left(6,\left\{[1,2],[2,5] \right\}\right)$}

Take
$$
\begin{array}{rcll}
P^I(N)&=&
\HD^{N+1}_{[1,2]}\HD^{N+1}_{[3,4]}\HD^{N}_{[3,6]}\HD^{N+1}_{[4,5]}
\left(C^7\right)&\mbox{(Figure \ref{F_lpos3a})}\\ 
P^{II}(N)&=&
\HD^{N+1}_{[1,5]}\HD^{N^2}_{[3,5]}\HD^{N}_{[5,6]}\left(C^7\right)
&\mbox{(Figure \ref{F_lpos3b}), and}\\   
P^{III}(N)&=&
\HD^{N+2}_{[1,2]}\HD^{N^2-N+2}_{[2,5]}\HD^{N}_{[1,6]}\left(C^7\right)&\mbox{(Figure \ref{F_lpos3c}).}\\
\end{array}
$$
Identify the elements of $P^{I}(N)$ with the elements of $P^{II}(N)$ at
ranks $1,2$, and $6$. 
Identify the elements of $P^{I}(N)$ with the elements of $P^{III}(N)$ at
rank $6$. Figure \ref{F_lpos3} represents the resulting poset for
$N=2$. 


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\caption{$P^I(N)$}
\label{F_lpos3a}
\end{figure}

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\caption{$P^{II}(N)$}
\label{F_lpos3b}
\end{figure}

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\end{center}
\caption{$P^{III}(N)$}
\label{F_lpos3c}
\end{figure}

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\special{ps: grestore}\end{picture}
\end{center}
\caption{$P\left(6,\left\{[1,2],[3,4],[4,5]\right\}\right)
+P\left(6,\left\{[3,5],[5,6]\right\}\right)
+P\left(6,\left\{[1,2],[2,5] \right\}\right)$}
\label{F_lpos3}
\end{figure}

\end{document}