diff --git a/school/di-ma/uebung/02/02_1.tex b/school/di-ma/uebung/02/02_1.tex index 6242ba6..9def870 100644 --- a/school/di-ma/uebung/02/02_1.tex +++ b/school/di-ma/uebung/02/02_1.tex @@ -44,6 +44,20 @@ Problem entspricht einer ungeordneten $k$-Mengenpartition, also $S_{n,k}$ + \begin{tabular}{c|cccccccccc} + $n/k$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\\hline + 0 & 1 &&&&&&&&& \\ + 1 & 0 & 1 & & & & & & & & \\ + 2 & 0 & 1 & 1 & & & & & & & \\ + 3 & 0 &1 &3 &1 & & & & & & \\ + 4 & 0 &1 &7 &6 &1 & & & & & \\ + 5 & 0 &1 &15 &25 &10 &1 & & & & \\ + 6 & 0 &1 &31 &90 &65 &15 &1 & & & \\ + 7 & 0 &1 &63 &301 &350 &140 &21 &1 & & \\ + 8 & 0 &1 &127 &966 &1701 &1050 &266 &28 &1 & \\ + 9 & 0 &1 &255 &3025 &7770 &\underline{6951} &2646 &462 &36 &1 \\ + \end{tabular} + \begin{eqnarray*} S_{n,k} &=& S_{n-1,k-1} + k * S_{n-1,k} \text{ mit} \\ S_{0,0} &=& 1 \\ @@ -73,64 +87,5 @@ \end{enumerate} -\section*{Aufgabe 2.2} -\begin{enumerate}[1.] - - \item Zyklenzerlegung: $(1\ 2\ 4) (3) (5\ 9) (6) (7\ 8)$ - - 2 Fixpunkte: $3$ und $6$ - - \item - \begin{eqnarray*} - s_{n.k} &=& s_{n-1,k-1} + (n-1)s_{n-1,k} \text{ mit} \\ - s_{0,0} &=& 1 \\ - s_{n,0} &=& 0 \\ - s_{n,n} &=& 1 \\\\ - s_{9,5} &=& 22449 - \end{eqnarray*} - -\end{enumerate} - -\section*{Aufgabe 2.3} -\begin{enumerate}[1.] - - \item - \begin{eqnarray*} - x_1 + x_2 + x_3 + x_4 + x_5 + x_6 &=& 67 \text{ mit } x_i \ge 0 \text{ für } 1 \le i \le 6 \\ - \text{Normalisierung:} \\ - \text{für } 1 \le i \le 3 \rightarrow x_i' &=& x_i - 1 \text{ (ungerade Zahlen werden gerade)} \\ - \text{für } 4 \le i \le 6 \rightarrow x_i' &=& x_i \\ - \Rightarrow x_1' + x_2' + x_3' + x_4' + x_5' + x_6' &=& 64 \\ - \text{für } 1 \le i \le 6 \rightarrow y_i &=& \frac{x_i'}{2} - \text{ (Bedingung \enquote{alle Zahlen gerade} erfüllt)} \\ - \Rightarrow y_1 + y_2 + y_3 + y_4 + y_5 + y_6 &=& 32 \\ - \text{für } 1 \le i \le 6 \rightarrow z_i &=& y_i + 1 \\ - \Rightarrow z_1 + z_2 + z_3 + z_4 + z_5 + y_6 &=& 38 \\\\ - \Rightarrow \binom{n-1}{k-1} &=& \binom{37}{5} = 435897 - \end{eqnarray*} - - \item - \begin{eqnarray*} - P_{n,k} &=& P_{n-1,k-1} + P_{n-k,k} \text{ mit} \\ - P_{n.0} &=& 0 \\ - P_{n.n} &=& 1 \\ - P_{n.1} &=& 1 \\\\ - P_{10,4} &=& 9 - \end{eqnarray*} - -\end{enumerate} - -\section*{Aufgabe 2.4} - -\begin{itemize} - - \item Nur Schritte nach rechts oder oben sind erlaubt - - \item Insgesamt $n$ Schritte nach oben und $k$ Schritte nach rechts - - \item $\Rightarrow \frac{(n + k)!}{n! k!} = \binom{n + k}{k}$ - -\end{itemize} - \end{document} diff --git a/school/di-ma/uebung/02/02_2.tex b/school/di-ma/uebung/02/02_2.tex new file mode 100644 index 0000000..b0728c3 --- /dev/null +++ b/school/di-ma/uebung/02/02_2.tex @@ -0,0 +1,73 @@ +\documentclass[12pt,a4paper,german]{article} +\usepackage{url} +%\usepackage{graphics} +\usepackage{times} +\usepackage[T1]{fontenc} +\usepackage{ngerman} +\usepackage{float} +\usepackage{diagbox} +\usepackage[utf8]{inputenc} +\usepackage{geometry} +\usepackage{amsfonts} +\usepackage{amsmath} +\usepackage{csquotes} +\usepackage{graphicx} +\usepackage{epsfig} +\usepackage{paralist} +\geometry{left=2.0cm,textwidth=17cm,top=3.5cm,textheight=23cm} + +%%%%%%%%%% Fill out the the definitions %%%%%%%%% +\def \name {Valentin Brandl} % +\def \matrikel {108018274494} % +\def \pname {Marvin Herrmann} % +\def \pmatrikel {108018265436} % +\def \gruppe {2 (Mi 10-12 Andre)} +\def \uebung {2} % +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + % DO NOT MODIFY THIS HEADER +\newcommand{\hwsol}{ +\vspace*{-2cm} +\noindent \matrikel \quad \name \hfill \"Ubungsgruppe: \gruppe \\ +\noindent \pmatrikel \quad \pname \\ +\begin{center}{\Large \bf L\"osung f\"ur \"Ubung \# \uebung}\end{center} +} + +\begin{document} +%Import header +\hwsol + +\section*{Aufgabe 2.2} +\begin{enumerate}[1.] + + \item Zyklenzerlegung: $(1\ 2\ 4) (3) (5\ 9) (6) (7\ 8)$ + + 2 Fixpunkte: $3$ und $6$ + + \item + \begin{tabular}{c|cccccccccc} + $n / k$ & 0&1&2&3&4&5&6&7&8&9 \\\hline + 0 & 1 &&&&&&&&& \\ + 1 & 0 & 1 &&&&&&&& \\ + 2 & 0 & 1 &1&&&&&&& \\ + 3 & 0 & 2 & 3 & 1 & & & & & & \\ + 4 & 0 & 6 & 11 & 6 & 1 & & & & & \\ + 5 & 0 & 24 & 50 & 35 & 10 & 1 & & & & \\ + 6 & 0 & 120 & 274 & 225 & 85 & 15 & 1 & & & \\ + 7 & 0 & 720 & 1764 & 1624 & 735 & 175 & 21 & 1 & & \\ + 8 & 0 & 5040 & 13068 & 13132 & 6769 & 1960 & 322 & 28 & 1 & \\ + 9 & 0 & 40320 & 109584 & 118124 & 67284 & \underline{22449} & 4536 & 546 & 36 & 1 \\ + \end{tabular} + + \begin{eqnarray*} + s_{n.k} &=& s_{n-1,k-1} + (n-1)s_{n-1,k} \text{ mit} \\ + s_{0,0} &=& 1 \\ + s_{n,0} &=& 0 \\ + s_{n,n} &=& 1 \\\\ + s_{9,5} &=& 22449 + \end{eqnarray*} + +\end{enumerate} + +\end{document} + diff --git a/school/di-ma/uebung/02/02_3.tex b/school/di-ma/uebung/02/02_3.tex new file mode 100644 index 0000000..0ac978a --- /dev/null +++ b/school/di-ma/uebung/02/02_3.tex @@ -0,0 +1,84 @@ +\documentclass[12pt,a4paper,german]{article} +\usepackage{url} +%\usepackage{graphics} +\usepackage{times} +\usepackage[T1]{fontenc} +\usepackage{ngerman} +\usepackage{float} +\usepackage{diagbox} +\usepackage[utf8]{inputenc} +\usepackage{geometry} +\usepackage{amsfonts} +\usepackage{amsmath} +\usepackage{csquotes} +\usepackage{graphicx} +\usepackage{epsfig} +\usepackage{paralist} +\geometry{left=2.0cm,textwidth=17cm,top=3.5cm,textheight=23cm} + +%%%%%%%%%% Fill out the the definitions %%%%%%%%% +\def \name {Valentin Brandl} % +\def \matrikel {108018274494} % +\def \pname {Marvin Herrmann} % +\def \pmatrikel {108018265436} % +\def \gruppe {2 (Mi 10-12 Andre)} +\def \uebung {2} % +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + % DO NOT MODIFY THIS HEADER +\newcommand{\hwsol}{ +\vspace*{-2cm} +\noindent \matrikel \quad \name \hfill \"Ubungsgruppe: \gruppe \\ +\noindent \pmatrikel \quad \pname \\ +\begin{center}{\Large \bf L\"osung f\"ur \"Ubung \# \uebung}\end{center} +} + +\begin{document} +%Import header +\hwsol + +\section*{Aufgabe 2.3} +\begin{enumerate}[1.] + + \item + \begin{eqnarray*} + x_1 + x_2 + x_3 + x_4 + x_5 + x_6 &=& 67 \text{ mit } x_i \ge 0 \text{ für } 1 \le i \le 6 \\ + \text{Normalisierung:} \\ + \text{für } 1 \le i \le 3 \rightarrow x_i' &=& x_i - 1 \text{ (ungerade Zahlen werden gerade)} \\ + \text{für } 4 \le i \le 6 \rightarrow x_i' &=& x_i \\ + \Rightarrow x_1' + x_2' + x_3' + x_4' + x_5' + x_6' &=& 64 \\ + \text{für } 1 \le i \le 6 \rightarrow y_i &=& \frac{x_i'}{2} + \text{ (Bedingung \enquote{alle Zahlen gerade} erfüllt)} \\ + \Rightarrow y_1 + y_2 + y_3 + y_4 + y_5 + y_6 &=& 32 \\ + \text{für } 1 \le i \le 6 \rightarrow z_i &=& y_i + 1 \\ + \Rightarrow z_1 + z_2 + z_3 + z_4 + z_5 + y_6 &=& 38 \\\\ + \Rightarrow \binom{n-1}{k-1} &=& \binom{37}{5} = 435897 + \end{eqnarray*} + + \item + \begin{tabular}{c|cccccccccc} + $n/k$ & 1 & 2 & 3 & 4 & 5 &6 & 7 & 8 & 9 & 10 \\\hline + 1 & 1 & & & & & & & & & \\ + 2 & 1 & 1 & & & & & & & & \\ + 3 & 1 & 1 & 1 & & & & & & & \\ + 4 & 1 & 2 & 1 & 1 & & & & & & \\ + 5 & 1 & 2 & 2 & 1 & 1 & & & & & \\ + 6 & 1 & 3 & 3 & 2 & 1 & 1 & & & & \\ + 7 & 1 & 3 & 4 & 3 & 2 & 1 & 1 & & & \\ + 8 & 1 & 4 & 5 & 5 & 3 & 2 & 1 & 1 & & \\ + 9 & 1 & 4 & 7 & 6 & 5 & 3 & 2 & 1 & 1 & \\ + 10 & 1 & 5 & 8 & \underline{9} &7 &5 &3 &2 & 1 & 1 \\ + \end{tabular} + + \begin{eqnarray*} + P_{n,k} &=& P_{n-1,k-1} + P_{n-k,k} \text{ mit} \\ + P_{n.0} &=& 0 \\ + P_{n.n} &=& 1 \\ + P_{n.1} &=& 1 \\\\ + P_{10,4} &=& 9 + \end{eqnarray*} + +\end{enumerate} + +\end{document} + diff --git a/school/di-ma/uebung/02/02_4.tex b/school/di-ma/uebung/02/02_4.tex new file mode 100644 index 0000000..f6ea366 --- /dev/null +++ b/school/di-ma/uebung/02/02_4.tex @@ -0,0 +1,53 @@ +\documentclass[12pt,a4paper,german]{article} +\usepackage{url} +%\usepackage{graphics} +\usepackage{times} +\usepackage[T1]{fontenc} +\usepackage{ngerman} +\usepackage{float} +\usepackage{diagbox} +\usepackage[utf8]{inputenc} +\usepackage{geometry} +\usepackage{amsfonts} +\usepackage{amsmath} +\usepackage{csquotes} +\usepackage{graphicx} +\usepackage{epsfig} +\usepackage{paralist} +\geometry{left=2.0cm,textwidth=17cm,top=3.5cm,textheight=23cm} + +%%%%%%%%%% Fill out the the definitions %%%%%%%%% +\def \name {Valentin Brandl} % +\def \matrikel {108018274494} % +\def \pname {Marvin Herrmann} % +\def \pmatrikel {108018265436} % +\def \gruppe {2 (Mi 10-12 Andre)} +\def \uebung {2} % +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + % DO NOT MODIFY THIS HEADER +\newcommand{\hwsol}{ +\vspace*{-2cm} +\noindent \matrikel \quad \name \hfill \"Ubungsgruppe: \gruppe \\ +\noindent \pmatrikel \quad \pname \\ +\begin{center}{\Large \bf L\"osung f\"ur \"Ubung \# \uebung}\end{center} +} + +\begin{document} +%Import header +\hwsol + +\section*{Aufgabe 2.4} + +\begin{itemize} + + \item Nur Schritte nach rechts oder oben sind erlaubt + + \item Insgesamt $n$ Schritte nach oben und $k$ Schritte nach rechts + + \item $\Rightarrow \frac{(n + k)!}{n! k!} = \binom{n + k}{k}$ + +\end{itemize} + +\end{document} +