\begin{table}%t4
\par
\caption{\label{table:dependencies}Do the properties of a population resulting from a certain pairing function depend on its generating properties? The three generating properties $f_{M}(M)$, $\fq$ and $\binf$ are listed in the top row. For each of the ten quantities listed in the left-most column, we list whether or not they depend on the choice of $f_{M}(M)$, $\fq$ and $\binf$. We indicate dependence and independence with the symbols $\surd$ and $\times$, respectively. The results for $\fqm$ and $\binfm$ are valid only for a population of binaries with identical primary mass $M_1$. The distribution over total system mass $f_{M_{\rm T}}$ includes both binary systems and single stars. 
  }
  \small%\centering
\par
  \begin{tabular}{ l | p{0.2cm}p{0.2cm}p{0.2cm}p{0.2cm} p{0.2cm}p{0.2cm}p{0.2cm}p{0.2cm} | p{0.2cm}p{0.2cm}p{0.2cm}p{0.2cm} p{0.2cm}p{0.2cm}p{0.2cm}p{0.2cm} | p{0.2cm}p{0.2cm}p{0.2cm}p{0.2cm} p{0.2cm}p{0.2cm}p{0.2cm}p{0.2cm} }
    \hline    \hline    
    Generating prop. & \multicolumn{8}{c}{$f_{M}(M)$}  & \multicolumn{8}{|c|}{$f_q(q)$}  & \multicolumn{8}{c}{$\binf$} \\ 
    \hline
    \multirow{0}{*}{Pairing function}   & 
    \multirow{0}{*}{ \rotatebox{-90}{\mbox{$\!\!\!\!$RP}}  }  & 
    \multirow{0}{*}{ \rotatebox{-90}{\mbox{$\!\!\!\!$PCRP}}  }  & 
    \multirow{0}{*}{ \rotatebox{-90}{\mbox{$\!\!\!\!$PCP-I}}  }  & 
    \multirow{0}{*}{ \rotatebox{-90}{\mbox{$\!\!\!\!$PCP-II}}  }  & 
    \multirow{0}{*}{ \rotatebox{-90}{\mbox{$\!\!\!\!$PCP-III}}  }  & 
    \multirow{0}{*}{ \rotatebox{-90}{\mbox{$\!\!\!\!$SCP-I}}  }  & 
    \multirow{0}{*}{ \rotatebox{-90}{\mbox{$\!\!\!\!$SCP-II}}  }  & 
    \multirow{0}{*}{ \rotatebox{-90}{\mbox{$\!\!\!\!$SCP-III}}  }  & 
    \multirow{0}{*}{ \rotatebox{-90}{\mbox{$\!\!\!\!$RP}}  }  & 
    \multirow{0}{*}{ \rotatebox{-90}{\mbox{$\!\!\!\!$PCRP}}  }  & 
    \multirow{0}{*}{ \rotatebox{-90}{\mbox{$\!\!\!\!$PCP-I}}  }  & 
    \multirow{0}{*}{ \rotatebox{-90}{\mbox{$\!\!\!\!${PCP-II}}} }  & 
    \multirow{0}{*}{ \rotatebox{-90}{\mbox{$\!\!\!\!$PCP-III}}  }  & 
    \multirow{0}{*}{ \rotatebox{-90}{\mbox{$\!\!\!\!$SCP-I}}  }  & 
    \multirow{0}{*}{ \rotatebox{-90}{\mbox{$\!\!\!\!$SCP-II}}  }  & 
    \multirow{0}{*}{ \rotatebox{-90}{\mbox{$\!\!\!\!$SCP-III}}  }  & 
    \multirow{0}{*}{ \rotatebox{-90}{\mbox{$\!\!\!\!$RP}}  }  & 
    \multirow{0}{*}{ \rotatebox{-90}{\mbox{$\!\!\!\!$PCRP}}  }  & 
    \multirow{0}{*}{ \rotatebox{-90}{\mbox{$\!\!\!\!$PCP-I}}  }  & 
    \multirow{0}{*}{ \rotatebox{-90}{\mbox{$\!\!\!\!$PCP-II}}  }  & 
    \multirow{0}{*}{ \rotatebox{-90}{\mbox{$\!\!\!\!$PCP-III}}  }  & 
    \multirow{0}{*}{ \rotatebox{-90}{\mbox{$\!\!\!\!$SCP-I}}  }  & 
    \multirow{0}{*}{ \rotatebox{-90}{\mbox{$\!\!\!\!$SCP-II}}  }  & 
    \multirow{0}{*}{ \rotatebox{-90}{\mbox{$\!\!\!\!$SCP-III}}  }  \\
       \hline
    $f_{M_1}$          & $\surd$  & $\surd$  & $\surd$  & $\surd$  & $\surd$  & $\surd$  & $\surd$  & $\surd$    & --       & --       & $\times$ & $\surd$  & $\times$ & $\surd$  & $\surd$  & $\surd$    & $\times$ & $\times$ & $\times$ & $\times$ & $\times$ & $\times$ & $\times$ & $\times$ \\
    $f_{M_2}$          & $\surd$  & $\surd$  & $\surd$  & $\surd$  & $\surd$  & $\surd$  & $\surd$  & $\surd$    & --       & --       & $\surd$  & $\surd$  & $\surd$  & $\surd$  & $\surd$  & $\surd$    & $\times$ & $\times$ & $\times$ & $\times$ & $\times$ & $\times$ & $\times$ & $\times$ \\
    $f_{M_{\rm T}}$          & $\surd$  & $\surd$  & $\surd$  & $\surd$  & $\surd$  & $\surd$  & $\surd$  & $\surd$    & --       & --       & $\surd$  & $\surd$  & $\surd$  & $\times$ & $\surd$  & $\times$   & $\surd$  & $\surd$  & $\surd$  & $\surd$  & $\surd$  & $\times$ & $\surd$  & $\times$ \\
    $f_{M_{1,S}}$      & $\surd$  & $\surd$  & $\surd$  & $\surd$  & $\surd$  & $\surd$  & $\surd$  & $\surd$    & --       & --       & $\times$ & $\times$ & $\times$ & $\surd$  & $\surd$  & $\surd$    & $\surd$  & $\times$ & $\times$ & $\times$ & $\times$ & $\surd$  & $\surd$  & $\surd$  \\
    $f_{M_{\rm S}}$          & $\surd$  & $\surd$  & $\surd$  & $\surd$  & $\surd$  & $\surd$  & $\surd$  & $\surd$    & --       & --       & $\times$ & $\surd$  & $\times$ & $\times$ & $\surd$  & $\times$   & $\times$ & $\times$ & $\times$ & $\surd$  & $\times$ & $\times$ & $\surd$  & $\times$ \\
    $f_{M_{\rm all}}$ (``IMF'')  & $\surd$  & $\surd$  & $\surd$  & $\surd$  & $\surd$  & $\surd$  & $\surd$  & $\surd$    & --       & --       & $\surd$  & $\surd$  & $\surd$  & $\surd$  & $\surd$  & $\surd$    & $\times$ & $\surd$  & $\surd$  & $\surd$  & $\surd$  & $\surd$  & $\surd$  & $\surd$  \\
    \hline
    $\fqall$           & $\surd$  & $\surd$  & $\times$ & $\surd$  & $\surd$  & $\times$ & $\surd$  & $\surd$    & --       & --       & $\surd$  & $\surd$  & $\surd$  & $\surd$  & $\surd$  & $\surd$    & $\times$ & $\times$ & $\times$ & $\times$ & $\times$ & $\times$ & $\times$ & $\times$ \\
    $\fqm$             & $\surd$  & $\surd$  & $\times$ & $\times$ & $\times$ & $\surd$  & $\surd$  & $\surd$    & --       & --       & $\surd$  & $\surd$  & $\surd$  & $\surd$  & $\surd$  & $\surd$    & $\times$ & $\times$ & $\times$ & $\times$ & $\times$ & $\times$ & $\times$ & $\times$ \\
    \hline
    $\binfall$         & $\times$ & $\times$ & $\times$ & $\surd$  & $\times$ & $\times$ & $\surd$  & $\times$   & --       & --       & $\times$ & $\surd$  & $\times$ & $\times$ & $\surd$  & $\times$   & $\surd$  & $\surd$  & $\surd$  & $\surd$  & $\surd$  & $\surd$  & $\surd$  & $\surd$  \\
    $\binfm$           & $\surd$  & $\times$ & $\times$ & $\times$ & $\times$ & $\surd$  & $\surd$  & $\surd$    & --       & --       & $\times$ & $\surd$  & $\times$ & $\surd$  & $\surd$  & $\surd$    & $\surd$  & $\surd$  & $\surd$  & $\surd$  & $\surd$  & $\surd$  & $\surd$  & $\surd$  \\
    \hline
  \end{tabular}
  \end{table}