diff options
Diffstat (limited to '')
| -rw-r--r-- | buch/chapters/50-permutationen/endlich.tex | 106 | ||||
| -rw-r--r-- | buch/chapters/50-permutationen/images/Makefile | 20 | ||||
| -rw-r--r-- | buch/chapters/50-permutationen/images/komposition.pdf | bin | 0 -> 13951 bytes | |||
| -rw-r--r-- | buch/chapters/50-permutationen/images/komposition.tex | 55 | ||||
| -rw-r--r-- | buch/chapters/50-permutationen/images/permutation.pdf | bin | 0 -> 13814 bytes | |||
| -rw-r--r-- | buch/chapters/50-permutationen/images/permutation.tex | 43 | ||||
| -rw-r--r-- | buch/chapters/50-permutationen/images/transpositionen.pdf | bin | 0 -> 22548 bytes | |||
| -rw-r--r-- | buch/chapters/50-permutationen/images/transpositionen.tex | 103 | ||||
| -rw-r--r-- | buch/chapters/50-permutationen/images/zyklenzerlegung.pdf | bin | 0 -> 14937 bytes | |||
| -rw-r--r-- | buch/chapters/50-permutationen/images/zyklenzerlegung.tex | 60 | ||||
| -rw-r--r-- | buch/chapters/50-permutationen/transpositionen.tex | 87 |
11 files changed, 285 insertions, 189 deletions
diff --git a/buch/chapters/50-permutationen/endlich.tex b/buch/chapters/50-permutationen/endlich.tex index 7669a17..700c0f2 100644 --- a/buch/chapters/50-permutationen/endlich.tex +++ b/buch/chapters/50-permutationen/endlich.tex @@ -21,31 +21,7 @@ Element} der Gruppe $S_n$ und wir auch mit $e$ bezeichnet. \subsection{Permutationen als $2\times n$-Matrizen} Eine Permutation kann als $2\times n$-Matrix geschrieben werden: \begin{center} -\begin{tikzpicture}[>=latex,thick] -\def\sx{0.8} -\def\sy{1} -\begin{scope}[xshift=-3cm] -\foreach \x in {1,...,6}{ - \node at ({(\x-1)*\sx},\sy) [above] {$\tiny\x$}; - \fill ({(\x-1)*\sx},\sy) circle[radius=0.05]; - \fill ({(\x-1)*\sx},0) circle[radius=0.05]; -} -\draw[->] (0,\sy) to[out=-70,in=110] (\sx,0); -\draw[<-] (0,0) to[out=70,in=-110] (\sx,\sy); -\draw[->] ({2*\sx},\sy) -- ({2*\sx},0); -\draw[->] ({3*\sx},\sy) to[out=-70,in=110] ({4*\sx},0); -\draw[->] ({4*\sx},\sy) to[out=-70,in=110] ({5*\sx},0); -\draw[->] ({5*\sx},\sy) to[out=-110,in=70] ({3*\sx},0); -\end{scope} -\node at (2.4,{\sy/2}) {$\mathstrut=\mathstrut$}; -\node at (5,{\sy/2}) {$\displaystyle -\renewcommand{\arraystretch}{1.4} -\begin{pmatrix} -1&2&3&4&5&6\\ -2&1&3&5&6&4 -\end{pmatrix} -$}; -\end{tikzpicture} +\includegraphics{chapters/50-permutationen/images/permutation.pdf} \end{center} Das neutrale Element hat die Matrix \[ @@ -64,43 +40,7 @@ Permutation angeordnet. Die zusammengesetzte Permutation kann dann in der zweiten Zeile der zweiten Permutation abgelesen werden: \begin{center} -\begin{tikzpicture}[>=latex,thick] -\begin{scope}[xshift=-4.5cm] -\node at (0,0) {$\displaystyle -\sigma_1=\begin{pmatrix} -1&2&3&4&5&6\\ -2&1&3&5&6&4 -\end{pmatrix}$}; -\node at (0,-1) {$\displaystyle -\sigma_2=\begin{pmatrix} -1&2&3&4&5&6\\ -3&4&5&6&1&2 -\end{pmatrix} -$}; -\end{scope} -\begin{scope} -\node at (0,0) {$\displaystyle -\begin{pmatrix} -1&2&3&4&5&6\\ -2&1&3&5&6&4 -\end{pmatrix}$}; -\node at (0,-1) {$\displaystyle -\begin{pmatrix} -2&1&3&5&6&4\\ -4&3&5&1&2&6 -\end{pmatrix} -$}; - -\end{scope} -\begin{scope}[xshift=4.5cm] -\node at (0,-0.5) {$\displaystyle -\sigma_2\sigma_1=\begin{pmatrix} -1&2&3&4&5&6\\ -4&3&5&1&2&6 -\end{pmatrix} -$}; -\end{scope} -\end{tikzpicture} +\includegraphics{chapters/50-permutationen/images/komposition.pdf} \end{center} Die Inverse einer Permutation kann erhalten werden, indem die beiden Zeilen vertauscht werden und dann die Spalten wieder so angeordnet werden, @@ -130,47 +70,7 @@ Eine Permutation $\sigma\in S_n$ kann auch mit sogenanten Zyklenzerlegung analysiert werden. Zum Beispiel: \begin{center} -\begin{tikzpicture}[>=latex,thick] -\begin{scope}[xshift=-3cm] -\node at (0,0) {$\displaystyle -\sigma=\begin{pmatrix} -{\color{red}1}&{\color{red}2}&{\color{darkgreen}3}&{\color{blue}4}&{\color{blue}5}&{\color{blue}6}\\ -{\color{red}2}&{\color{red}1}&{\color{darkgreen}3}&{\color{blue}5}&{\color{blue}6}&{\color{blue}4} -\end{pmatrix}$}; -\end{scope} -\node at (0,0) {$\mathstrut=\mathstrut$}; -\begin{scope}[xshift=1.5cm] -\coordinate (A) at (0,0.5); -\coordinate (B) at (0,-0.5); -\draw[->,color=red] (A) to[out=-20,in=20] (0,-0.5); -\draw[->,color=red] (B) to[out=160,in=-160] (0,0.5); -\node at (A) [above] {$\tiny 1$}; -\node at (B) [below] {$\tiny 2$}; -\fill (A) circle[radius=0.05]; -\fill (B) circle[radius=0.05]; - -\coordinate (C) at (1.5,0.25); -\node at (C) [above] {$\tiny 3$}; -\draw[->,color=darkgreen] ({1.5+0.01},0.25) to[out=-10,in=-170] ({1.5-0.01},0.25); -\draw[color=darkgreen] (1.5,{0.25-0.3}) circle[radius=0.3]; -\fill (C) circle[radius=0.05]; - -\def\r{0.5} -\coordinate (D) at ({3.5+\r*cos(90)},{0+\r*sin(90)}); -\coordinate (E) at ({3.5+\r*cos(210)},{0+\r*sin(210)}); -\coordinate (F) at ({3.5+\r*cos(330)},{0+\r*sin(330)}); -\node at (D) [above] {$\tiny 4$}; -\node at (E) [below left] {$\tiny 5$}; -\node at (F) [below right] {$\tiny 6$}; -\draw[->,color=blue] (D) to[out=180,in=120] (E); -\draw[->,color=blue] (E) to[out=-60,in=-120] (F); -\draw[->,color=blue] (F) to[out=60,in=0] (D); -\fill (D) circle[radius=0.05]; -\fill (E) circle[radius=0.05]; -\fill (F) circle[radius=0.05]; - -\end{scope} -\end{tikzpicture} +\includegraphics{chapters/50-permutationen/images/zyklenzerlegung.pdf} \end{center} \begin{definition} diff --git a/buch/chapters/50-permutationen/images/Makefile b/buch/chapters/50-permutationen/images/Makefile new file mode 100644 index 0000000..c2e0e9f --- /dev/null +++ b/buch/chapters/50-permutationen/images/Makefile @@ -0,0 +1,20 @@ +# +# Makefile -- Bilder +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +all: transpositionen.pdf zyklenzerlegung.pdf permutation.pdf \ + komposition.pdf + +permutation.pdf: permutation.tex + pdflatex permutation.tex + +komposition.pdf: komposition.tex + pdflatex komposition.tex + +transpositionen.pdf: transpositionen.tex + pdflatex transpositionen.tex + +zyklenzerlegung.pdf: zyklenzerlegung.tex + pdflatex zyklenzerlegung.tex + diff --git a/buch/chapters/50-permutationen/images/komposition.pdf b/buch/chapters/50-permutationen/images/komposition.pdf Binary files differnew file mode 100644 index 0000000..9e532bf --- /dev/null +++ b/buch/chapters/50-permutationen/images/komposition.pdf diff --git a/buch/chapters/50-permutationen/images/komposition.tex b/buch/chapters/50-permutationen/images/komposition.tex new file mode 100644 index 0000000..ef3ec65 --- /dev/null +++ b/buch/chapters/50-permutationen/images/komposition.tex @@ -0,0 +1,55 @@ +% +% komposition.tex -- Komposition zweier Permutationen +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math} +\begin{document} +\def\skala{1} +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\begin{scope}[xshift=-4.5cm] +\node at (0,0) {$\displaystyle +\sigma_1=\begin{pmatrix} +1&2&3&4&5&6\\ +2&1&3&5&6&4 +\end{pmatrix}$}; +\node at (0,-1) {$\displaystyle +\sigma_2=\begin{pmatrix} +1&2&3&4&5&6\\ +3&4&5&6&1&2 +\end{pmatrix} +$}; +\end{scope} +\begin{scope} +\node at (0,0) {$\displaystyle +\begin{pmatrix} +1&2&3&4&5&6\\ +2&1&3&5&6&4 +\end{pmatrix}$}; +\node at (0,-1) {$\displaystyle +\begin{pmatrix} +2&1&3&5&6&4\\ +4&3&5&1&2&6 +\end{pmatrix} +$}; + +\end{scope} +\begin{scope}[xshift=4.5cm] +\node at (0,-0.5) {$\displaystyle +\sigma_2\sigma_1=\begin{pmatrix} +1&2&3&4&5&6\\ +4&3&5&1&2&6 +\end{pmatrix} +$}; +\end{scope} + +\end{tikzpicture} +\end{document} + diff --git a/buch/chapters/50-permutationen/images/permutation.pdf b/buch/chapters/50-permutationen/images/permutation.pdf Binary files differnew file mode 100644 index 0000000..cdfa186 --- /dev/null +++ b/buch/chapters/50-permutationen/images/permutation.pdf diff --git a/buch/chapters/50-permutationen/images/permutation.tex b/buch/chapters/50-permutationen/images/permutation.tex new file mode 100644 index 0000000..ee58d4a --- /dev/null +++ b/buch/chapters/50-permutationen/images/permutation.tex @@ -0,0 +1,43 @@ +% +% permutation.tex -- Definition einer Permutation +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math} +\begin{document} +\def\skala{1} +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\def\sx{0.8} +\def\sy{1} +\begin{scope}[xshift=-3cm] +\foreach \x in {1,...,6}{ + \node at ({(\x-1)*\sx},\sy) [above] {$\tiny\x$}; + \fill ({(\x-1)*\sx},\sy) circle[radius=0.05]; + \fill ({(\x-1)*\sx},0) circle[radius=0.05]; +} +\draw[->] (0,\sy) to[out=-70,in=110] (\sx,0); +\draw[<-] (0,0) to[out=70,in=-110] (\sx,\sy); +\draw[->] ({2*\sx},\sy) -- ({2*\sx},0); +\draw[->] ({3*\sx},\sy) to[out=-70,in=110] ({4*\sx},0); +\draw[->] ({4*\sx},\sy) to[out=-70,in=110] ({5*\sx},0); +\draw[->] ({5*\sx},\sy) to[out=-110,in=70] ({3*\sx},0); +\end{scope} +\node at (2.4,{\sy/2}) {$\mathstrut=\mathstrut$}; +\node at (5,{\sy/2}) {$\displaystyle +\renewcommand{\arraystretch}{1.4} +\begin{pmatrix} +1&2&3&4&5&6\\ +2&1&3&5&6&4 +\end{pmatrix} +$}; + +\end{tikzpicture} +\end{document} + diff --git a/buch/chapters/50-permutationen/images/transpositionen.pdf b/buch/chapters/50-permutationen/images/transpositionen.pdf Binary files differnew file mode 100644 index 0000000..ad2ea7c --- /dev/null +++ b/buch/chapters/50-permutationen/images/transpositionen.pdf diff --git a/buch/chapters/50-permutationen/images/transpositionen.tex b/buch/chapters/50-permutationen/images/transpositionen.tex new file mode 100644 index 0000000..0670a2c --- /dev/null +++ b/buch/chapters/50-permutationen/images/transpositionen.tex @@ -0,0 +1,103 @@ +% +% transpositionen.tex -- Darstellung eines Zyklus aus Transpositionen +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math} +\begin{document} +\def\skala{1} + +\def\kreuz#1#2#3{ + \draw[->] ({(#1)-1},#2) to[out=-90,in=90] ({#1},{#2-1}); + \draw[->] ({#1},#2) to[out=-90,in=90] ({#1-1},{#2-1}); + \node at ({(#1)-0.5+0.2},{#2-0.5}) [right] {$#3$}; +} +\begin{tikzpicture}[>=latex,thick,scale=\skala] +\foreach \x in {1,2,3,6,7,8,9}{ + \fill ({\x-1},0) circle[radius=0.05]; +} +\foreach \x in {1,2,3}{ + \node at ({\x-1},0) [above] {$\tiny \x$}; +} +\node at (8,0) [above] {$\tiny k$}; +\node at (7,0) [above] {$\tiny k-1$}; +\node at (6,0) [above] {$\tiny k-2$}; +\node at (5,0) [above] {$\tiny k-3$}; +\foreach \x in {1,2,3,4,7,8,9}{ + \fill ({\x-1},-8) circle[radius=0.05]; +} +\foreach \x in {1,2,3,4}{ + \node at ({\x-1},-8) [below] {$\tiny \x$}; +} +\node at (6,-8) [below] {$k-2$}; +\node at (7,-8) [below] {$k-1$}; +\node at (8,-8) [below] {$k$}; + +\foreach \x in {3,3.2,...,5}{ + \fill (\x,{-8+\x}) circle[radius=0.02]; + \fill ({\x+0.5},-8) circle[radius=0.02]; + \fill ({\x-0.5},0) circle[radius=0.02]; +} + +\kreuz{8}{0}{\tau_{k-1,k}} +\kreuz{7}{-1}{\tau_{k-2,k-1}} +\kreuz{6}{-2}{\tau_{k-3,k-2}} +%\kreuz{5}{-3}{\tau_{56}} +%\kreuz{4}{-4}{\tau_{45}} +\kreuz{3}{-5}{\tau_{34}} +\kreuz{2}{-6}{\tau_{23}} +\kreuz{1}{-7}{\tau_{12}} + +\draw[->,color=gray] (0,0) -- (0,-7); +\draw[->,color=gray] (1,0) -- (1,-6); +\draw[->,color=gray] (2,0) -- (2,-5); +%\draw[->,color=gray] (3,0) -- (3,-4); +%\draw[->,color=gray] (4,0) -- (4,-3); +\draw[->,color=gray] (5,0) -- (5,-2); +\draw[->,color=gray] (6,0) -- (6,-1); + +\draw[->,color=gray] (8,-1) -- (8,-8); +\draw[->,color=gray] (7,-2) -- (7,-8); +\draw[->,color=gray] (6,-3) -- (6,-8); +%\draw[->,color=gray] (5,-4) -- (5,-8); +%\draw[->,color=gray] (4,-5) -- (4,-8); +\draw[->,color=gray] (3,-6) -- (3,-8); +\draw[->,color=gray] (2,-7) -- (2,-8); + +\fill (6,-1) circle[radius=0.05]; +\fill (7,-1) circle[radius=0.05]; +\fill (8,-1) circle[radius=0.05]; + +\fill (5,-2) circle[radius=0.05]; +\fill (6,-2) circle[radius=0.05]; +\fill (7,-2) circle[radius=0.05]; + +%\fill (4,-3) circle[radius=0.05]; +\fill (5,-3) circle[radius=0.05]; +\fill (6,-3) circle[radius=0.05]; + +%\fill (3,-4) circle[radius=0.05]; +%\fill (4,-4) circle[radius=0.05]; +%\fill (5,-4) circle[radius=0.05]; + +\fill (2,-5) circle[radius=0.05]; +\fill (3,-5) circle[radius=0.05]; +%\fill (4,-5) circle[radius=0.05]; + +\fill (1,-6) circle[radius=0.05]; +\fill (2,-6) circle[radius=0.05]; +\fill (3,-6) circle[radius=0.05]; + +\fill (0,-7) circle[radius=0.05]; +\fill (1,-7) circle[radius=0.05]; +\fill (2,-7) circle[radius=0.05]; + +\end{tikzpicture} +\end{document} + diff --git a/buch/chapters/50-permutationen/images/zyklenzerlegung.pdf b/buch/chapters/50-permutationen/images/zyklenzerlegung.pdf Binary files differnew file mode 100644 index 0000000..72d9353 --- /dev/null +++ b/buch/chapters/50-permutationen/images/zyklenzerlegung.pdf diff --git a/buch/chapters/50-permutationen/images/zyklenzerlegung.tex b/buch/chapters/50-permutationen/images/zyklenzerlegung.tex new file mode 100644 index 0000000..c197579 --- /dev/null +++ b/buch/chapters/50-permutationen/images/zyklenzerlegung.tex @@ -0,0 +1,60 @@ +% +% zyklenzerlegung.tex -- Zerlegung einer Permutation in Zyklen +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math} +\begin{document} +\definecolor{darkgreen}{rgb}{0,0.6,0} +\def\skala{1} +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\begin{scope}[xshift=-3cm] +\node at (0,0) {$\displaystyle +\sigma=\begin{pmatrix} +{\color{red}1}&{\color{red}2}&{\color{darkgreen}3}&{\color{blue}4}&{\color{blue}5}&{\color{blue}6}\\ +{\color{red}2}&{\color{red}1}&{\color{darkgreen}3}&{\color{blue}5}&{\color{blue}6}&{\color{blue}4} +\end{pmatrix}$}; +\end{scope} +\node at (0,0) {$\mathstrut=\mathstrut$}; +\begin{scope}[xshift=1.5cm] +\coordinate (A) at (0,0.5); +\coordinate (B) at (0,-0.5); +\draw[->,color=red] (A) to[out=-20,in=20] (0,-0.5); +\draw[->,color=red] (B) to[out=160,in=-160] (0,0.5); +\node at (A) [above] {$\tiny 1$}; +\node at (B) [below] {$\tiny 2$}; +\fill (A) circle[radius=0.05]; +\fill (B) circle[radius=0.05]; + +\coordinate (C) at (1.5,0.25); +\node at (C) [above] {$\tiny 3$}; +\draw[->,color=darkgreen] ({1.5+0.01},0.25) to[out=-10,in=-170] ({1.5-0.01},0.25); +\draw[color=darkgreen] (1.5,{0.25-0.3}) circle[radius=0.3]; +\fill (C) circle[radius=0.05]; + +\def\r{0.5} +\coordinate (D) at ({3.5+\r*cos(90)},{0+\r*sin(90)}); +\coordinate (E) at ({3.5+\r*cos(210)},{0+\r*sin(210)}); +\coordinate (F) at ({3.5+\r*cos(330)},{0+\r*sin(330)}); +\node at (D) [above] {$\tiny 4$}; +\node at (E) [below left] {$\tiny 5$}; +\node at (F) [below right] {$\tiny 6$}; +\draw[->,color=blue] (D) to[out=180,in=120] (E); +\draw[->,color=blue] (E) to[out=-60,in=-120] (F); +\draw[->,color=blue] (F) to[out=60,in=0] (D); +\fill (D) circle[radius=0.05]; +\fill (E) circle[radius=0.05]; +\fill (F) circle[radius=0.05]; + +\end{scope} + +\end{tikzpicture} +\end{document} + diff --git a/buch/chapters/50-permutationen/transpositionen.tex b/buch/chapters/50-permutationen/transpositionen.tex index baed2fb..604e010 100644 --- a/buch/chapters/50-permutationen/transpositionen.tex +++ b/buch/chapters/50-permutationen/transpositionen.tex @@ -35,92 +35,7 @@ also die Permutation, die $1\to2\to3\to\dots\to k-2\to k-1\to k\to 1$ abbildet. Dieser Zyklus lässt sich wie folgt aus Transpositionen zusammensetzen: \begin{center} -\def\kreuz#1#2#3{ - \draw[->] ({(#1)-1},#2) to[out=-90,in=90] ({#1},{#2-1}); - \draw[->] ({#1},#2) to[out=-90,in=90] ({#1-1},{#2-1}); - \node at ({(#1)-0.5+0.2},{#2-0.5}) [right] {$#3$}; -} -\begin{tikzpicture}[>=latex,thick] -\foreach \x in {1,2,3,6,7,8,9}{ - \fill ({\x-1},0) circle[radius=0.05]; -} -\foreach \x in {1,2,3}{ - \node at ({\x-1},0) [above] {$\tiny \x$}; -} -\node at (8,0) [above] {$\tiny k$}; -\node at (7,0) [above] {$\tiny k-1$}; -\node at (6,0) [above] {$\tiny k-2$}; -\node at (5,0) [above] {$\tiny k-3$}; -\foreach \x in {1,2,3,4,7,8,9}{ - \fill ({\x-1},-8) circle[radius=0.05]; -} -\foreach \x in {1,2,3,4}{ - \node at ({\x-1},-8) [below] {$\tiny \x$}; -} -\node at (6,-8) [below] {$k-2$}; -\node at (7,-8) [below] {$k-1$}; -\node at (8,-8) [below] {$k$}; - -\foreach \x in {3,3.2,...,5}{ - \fill (\x,{-8+\x}) circle[radius=0.02]; - \fill ({\x+0.5},-8) circle[radius=0.02]; - \fill ({\x-0.5},0) circle[radius=0.02]; -} - -\kreuz{8}{0}{\tau_{k-1,k}} -\kreuz{7}{-1}{\tau_{k-2,k-1}} -\kreuz{6}{-2}{\tau_{k-3,k-2}} -%\kreuz{5}{-3}{\tau_{56}} -%\kreuz{4}{-4}{\tau_{45}} -\kreuz{3}{-5}{\tau_{34}} -\kreuz{2}{-6}{\tau_{23}} -\kreuz{1}{-7}{\tau_{12}} - -\draw[->,color=gray] (0,0) -- (0,-7); -\draw[->,color=gray] (1,0) -- (1,-6); -\draw[->,color=gray] (2,0) -- (2,-5); -%\draw[->,color=gray] (3,0) -- (3,-4); -%\draw[->,color=gray] (4,0) -- (4,-3); -\draw[->,color=gray] (5,0) -- (5,-2); -\draw[->,color=gray] (6,0) -- (6,-1); - -\draw[->,color=gray] (8,-1) -- (8,-8); -\draw[->,color=gray] (7,-2) -- (7,-8); -\draw[->,color=gray] (6,-3) -- (6,-8); -%\draw[->,color=gray] (5,-4) -- (5,-8); -%\draw[->,color=gray] (4,-5) -- (4,-8); -\draw[->,color=gray] (3,-6) -- (3,-8); -\draw[->,color=gray] (2,-7) -- (2,-8); - -\fill (6,-1) circle[radius=0.05]; -\fill (7,-1) circle[radius=0.05]; -\fill (8,-1) circle[radius=0.05]; - -\fill (5,-2) circle[radius=0.05]; -\fill (6,-2) circle[radius=0.05]; -\fill (7,-2) circle[radius=0.05]; - -%\fill (4,-3) circle[radius=0.05]; -\fill (5,-3) circle[radius=0.05]; -\fill (6,-3) circle[radius=0.05]; - -%\fill (3,-4) circle[radius=0.05]; -%\fill (4,-4) circle[radius=0.05]; -%\fill (5,-4) circle[radius=0.05]; - -\fill (2,-5) circle[radius=0.05]; -\fill (3,-5) circle[radius=0.05]; -%\fill (4,-5) circle[radius=0.05]; - -\fill (1,-6) circle[radius=0.05]; -\fill (2,-6) circle[radius=0.05]; -\fill (3,-6) circle[radius=0.05]; - -\fill (0,-7) circle[radius=0.05]; -\fill (1,-7) circle[radius=0.05]; -\fill (2,-7) circle[radius=0.05]; - -\end{tikzpicture} +\includegraphics{chapters/50-permutationen/images/transpositionen.pdf} \end{center} Es ist also \[ |