The Main Counting Theorems.tex

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\textbf{}%
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\multicolumn{5}{|c|}{\textbf{The fundamental lemma of labeled counting}}\tabularnewline
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$\mathcal{F}$ &  & $\mathcal{F}'$ &  & $\mathcal{F}''$\tabularnewline
$\cdots$
\Deck[0.7]{r}{$S_r$}{$d_r$}{ \Rabbit[0.7] }
$\cdots$ & $=$ & $\cdots$
\Deck[0.7]{r}{$S_r$}{$d'_r$}{ \Rabbit[0.7] }
$\cdots$ & $\oplus$ & $\cdots$
\Deck[0.7]{r}{$S_r$}{$d''_r$}{ \Rabbit[0.7] }
$\cdots$\tabularnewline
$d_{r}=d'_{r}+d''_{r}$ &  &  &  & \tabularnewline
 &  & $\Downarrow$ &  & \tabularnewline
 &  &  &  & \tabularnewline
\multicolumn{5}{|c|}{$\mathcal{H}\left(x,y\right)=\mathcal{H}'\left(x,y\right)\mathcal{H}''\left(x,y\right)$}\tabularnewline
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\textbf{The exponential formula (in a 'Sorcerer's Apprentice' fashion)}\tabularnewline
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\textbf{The Trickle} \tabularnewline
\Deck[0.7]{}{}{0}{}
\Deck[0.7]{}{}{0}{}
\Deck[0.7]{}{}{0}{}
$\cdots$
\Deck[0.7]{r}{\{$\cdots$\}}{1}{ \Rabbit[0.7] }
$\cdots$\tabularnewline
$\downarrow\oplus$\tabularnewline
\textbf{The Flow} \tabularnewline
\Deck[0.7]{}{}{0}{}
\Deck[0.7]{}{}{0}{}
\Deck[0.7]{}{}{0}{}
$\cdots$
\Deck[0.7]{r}{\{$\cdots$\}}{$d_r$}{ \Rabbit[0.7] }
$\cdots$\tabularnewline
$\downarrow\oplus$\tabularnewline
\textbf{The Flood} \tabularnewline
\Deck[0.7]{1}{\{$\cdots$\}}{$d_1$}{ \Rabbit[0.7] }
\Deck[0.7]{2}{\{$\cdots$\}}{$d_2$}{ \Rabbit[0.7] }
\Deck[0.7]{3}{\{$\cdots$\}}{$d_3$}{ \Rabbit[0.7] }
$\cdots$
\Deck[0.7]{r}{\{$\cdots$\}}{$d_r$}{ \Rabbit[0.7] }
$\cdots$\tabularnewline
\tabularnewline
\textbf{$\Downarrow$}\tabularnewline
\tabularnewline
$\mathcal{H}\left(x,y\right)=e^{y\mathcal{D}\left(x\right)}$\tabularnewline
\tabularnewline
$nh_{n}=\sum_{k}\binom{n}{k}kd_{k}h_{n-k}\qquad\left(n\geq1;h_{0}=1\right)$\tabularnewline
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