%hide
%latex
\title{Hierachical Tuple}
\author{Patrick Hammer}
\date{September 7, 2011}
\maketitle
\section{Preamble}
This document describes some mathematical structure I needed for AI and linguistic purposes.
\section{Hierachical Tuple}
Let $T$ be a tuple. $H(T)$, the hierachical tuple of the tuple $T$, is the tuple, which includes the (tuples of same cardinality sorted like in $T$), of the specific subsets of the
powerset of $T$, which elements are neighbours in $T$,
(again sorted like in $T$), (longer tuples at first) and doesn't include the empty set.
\subsection{Cardinality}
Let $H$ be a hierachical tuple of $T$, let's write $H(T)$, then:
\begin{equation}
|H(T)|=\frac{|T|*(|T|+1)}{2}
\end{equation}
\subsection{Proof}
There are $|T|-(m-1)$ possibilities, to create a tuple with $m$ elements of $T$,
which are neighbours in $T$.
That implies:
\begin{equation}
|H(T)|=\sum_{m=1}^{|T|}|T|-(m-1)=\sum_{m=1}^{|T|}m=\frac{|T|*(|T|+1)}{2}
\end{equation}
\subsection{Example}
\begin{equation}
\nonumber $T=(a,b,c)$
\end{equation}
\begin{equation}
\nonumber $P(T)=\{\{a,b,c\},\{a,b\},\{b,c\},\{a,c\},\{a\},\{b\},\{c\},\{\}\}$
\end{equation}
\begin{equation}
\nonumber $H(T)=((a,b,c),(a,b),(b,c),(a),(b),(c))$
\end{equation}
