From 71963a0e396b6781d39bf9c3dfe2e76e44d9f5a2 Mon Sep 17 00:00:00 2001
From: Paul Jungeblut <paul.jungeblut@gmail.com>
Date: Sun, 14 Feb 2016 01:46:45 +0100
Subject: Improving math section.

---
 math/math.tex | 55 +++++++++++++++++++++++++++++++++++++++++++++----------
 1 file changed, 45 insertions(+), 10 deletions(-)

(limited to 'math/math.tex')

diff --git a/math/math.tex b/math/math.tex
index 7093f78..51410bd 100644
--- a/math/math.tex
+++ b/math/math.tex
@@ -66,20 +66,55 @@ Multipliziert Polynome $A$ und $B$.
 \subsection{Kombinatorik}
 
 \subsubsection{Berühmte Zahlen}
-\begin{tabular}{|l|l|l|}
+\begin{tabularx}{\textwidth}{|l|X|l|}
 	\hline
-	\textsc{Fibonacci}-Zahlen	& $f(0) = 0 \quad f(1) = 1 \quad f(n+2) = f(n+1) + f(n)$	& Bem. \ref{bem:fibonacciMat}, \ref{bem:fibonacciGreedy}\\
-	\textsc{Catalan}-Zahlen		& $C_0 = 1 \quad C_n = \sum\limits_{k = 0}^{n - 1} C_kC_{n - 1 - k} = \frac{1}{n + 1}{2n \choose n} = \frac{2(2n - 1)}{n+1} \cdot C_{n-1}$	& Bem. \ref{bem:catalanOverflow}, \ref{bem:catalanAnwendung}\\
-	\textsc{Euler}-Zahlen (I)	& $\left\langle\begin{array}{c} n \\ 0\end{array}\right\rangle = \left\langle\begin{array}{c} n \\ n-1 \end{array}\right\rangle = 1 \quad \left\langle\begin{array}{c} n \\ k\end{array}\right\rangle = (k + 1)\left\langle\begin{array}{c} n-1 \\ k\end{array}\right\rangle + (n-k)\left\langle\begin{array}{c} n-1 \\ k-1\end{array}\right\rangle$ & Bem. \ref{bem:euler1}\\
-	\textsc{Euler}-Zahlen (II)	& $\left\langle\left\langle\begin{array}{c}n\\0\end{array}\right\rangle\right\rangle = 1 \quad \left\langle\left\langle\begin{array}{c}n\\n\end{array}\right\rangle\right\rangle = 0 \quad \left\langle\left\langle\begin{array}{c}n\\k\end{array}\right\rangle\right\rangle = (k + 1)\left\langle\left\langle\begin{array}{c}n-1\\k\end{array}\right\rangle\right\rangle + (2n - k - 1)\left\langle\left\langle\begin{array}{c}n-1\\k-1\end{array}\right\rangle\right\rangle$ & Bem. \ref{bem:euler2}\\
-	\textsc{Stirling}-Zahlen (I)	& $\left[\begin{array}{c}0\\0\end{array}\right] = 1 \quad \left[\begin{array}{c}n\\0\end{array}\right] = \left[\begin{array}{c}0\\n\end{array}\right] = 0 \quad \left[\begin{array}{c}n\\k\end{array}\right] = \left[\begin{array}{c}n-1\\k-1\end{array}\right] + (n-1)\left[\begin{array}{c}n-1\\k\end{array}\right]$ & Bem. \ref{bem:stirling1}\\
-	\textsc{Stirling}-Zahlen (II)	& $\left\{\begin{array}{c}n\\1\end{array}\right\} = \left\{\begin{array}{c}n\\n\end{array}\right\} = 1 \quad \left\{\begin{array}{c}n\\k\end{array}\right\} = k\left\{\begin{array}{c}n-1\\k\end{array}\right\} + \left\{\begin{array}{c}n-1\\k-1\end{array}\right\}$ & Bem. \ref{bem:stirling2}\\
-	Integer-Partitions		& $f(1,1) = 1 \quad f(n,k) = 0 \text{ für } k > n \quad f(n,k)  = f(n-k,k) + f(n,k-1)$ & Bem. \ref{bem:integerPartitions}\\
+	\textsc{Fibonacci}-Zahlen	&
+	$f(0) = 0 \qquad
+	f(1) = 1 \qquad
+	f(n+2) = f(n+1) + f(n)$ &
+	Bem. \ref{bem:fibonacciMat}, \ref{bem:fibonacciGreedy} \\
+
+	\textsc{Catalan}-Zahlen	&
+	$C_0 = 1 \qquad
+	C_n = \sum\limits_{k = 0}^{n - 1} C_kC_{n - 1 - k} = \frac{1}{n + 1}\binom{2n}{n} = \frac{2(2n - 1)}{n+1} \cdot C_{n-1}$ &
+	Bem. \ref{bem:catalanOverflow}, \ref{bem:catalanAnwendung} \\
+
+	\textsc{Euler}-Zahlen (I) &
+	$\eulerI{n}{0} = \eulerI{n}{n-1} = 1 \qquad
+	\eulerI{n}{k} = (k+1) \eulerI{n-1}{k} + (n-k) \eulerI{n-1}{k-1} $ &
+	Bem. \ref{bem:euler1} \\
+
+	\textsc{Euler}-Zahlen (II) &
+	$\eulerII{n}{0} = 1 \qquad
+	\eulerII{n}{n} = 0 \qquad
+	\eulerII{n}{k} = (k+1) \eulerII{n-1}{k} + (2n-k-1) \eulerII{n-1}{k-1}$ &
+	Bem. \ref{bem:euler2} \\
+
+	\textsc{Stirling}-Zahlen (I) &
+	$\stirlingI{0}{0} = 1 \qquad
+	\stirlingI{n}{0} = \stirlingI{0}{n} = 0 \qquad
+	\stirlingI{n}{k} = \stirlingI{n-1}{k-1} + (n-1) \stirlingI{n-1}{k}$ &
+	Bem. \ref{bem:stirling1} \\
+
+	\textsc{Stirling}-Zahlen (II) &
+	$\stirlingII{n}{1} = \stirlingII{n}{n} = 1 \qquad
+	\stirlingII{n}{k} = k \stirlingII{n-1}{k} + \stirlingII{n-1}{k-1}$ &
+	Bem. \ref{bem:stirling2} \\
+
+	Integer-Partitions &
+	$f(1,1) = 1 \qquad f(n,k) = 0 \text{ für } k > n \qquad f(n,k)  = f(n-k,k) + f(n,k-1)$ &
+	Bem. \ref{bem:integerPartitions} \\
 	\hline
-\end{tabular}
+\end{tabularx}
 
 \begin{bem}\label{bem:fibonacciMat}
-$\left(\begin{array}{cc} 0 & 1 \\ 1 & 1\end{array}\right)^n \cdot \left(\begin{array}{c}0 \\ 1\end{array}\right) = \left(\begin{array}{c}f_n \\ f_{n+1}\end{array}\right)$
+$
+\begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix}^n
+\cdot
+\begin{pmatrix} 0 \\ 1 \end{pmatrix}
+=
+\begin{pmatrix}f_n \\ f_{n+1} \end{pmatrix}
+$
 \end{bem}
 
 \begin{bem}[\textsc{Zeckendorfs} Theorem]\label{bem:fibonacciGreedy}
-- 
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