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annotate notes/tex/combinatorics.tex @ 45:e65f4b1a6e32
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author | Markus Kaiser <markus.kaiser@in.tum.de> |
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date | Wed, 08 Jan 2014 14:26:02 +0100 |
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39 | 1 \defineUnit{zaehlen}{% |
2 \begin{frame} | |
3 \frametitle{Faktorielle} | |
4 | |
5 \begin{definition}[Fakultät] | |
6 Die \structure{Fakultät $n!$} einer natürlichen Zahl $n \in \N_0$ ist | |
7 \[ n! \defeq \prod_{i=1}^n i = n \cdot (n - 1) \cdot \ldots \cdot 1 \] | |
8 mit $0! \defeq 1$. | |
9 \end{definition} | |
10 | |
11 \vfill | |
12 | |
13 \begin{definition}[Steigende und fallende Faktorielle] | |
14 Für $n, m \in \N_0$ mit $m \leq n$ ist | |
15 { | |
16 \setlength{\belowdisplayskip}{0pt} | |
17 \begin{align} | |
18 n^{\underline m} &\defeq \frac{n!}{(n-m)!} \tag{\structure{fallende Faktorielle}}\\ | |
19 &= n \cdot (n - 1) \cdot \ldots \cdot (n - m + 1) \\ | |
20 \intertext{\vspace{1em}} | |
21 n^{\overline m} &\defeq \frac{(n+m-1)!}{(n-1)!} \tag{\structure{steigende Faktorielle}}\\ | |
22 &= n \cdot (n + 1) \cdot \ldots \cdot (n + m - 1) | |
23 \end{align} | |
24 } | |
25 \end{definition} | |
26 \end{frame} | |
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27 } |
39 | 28 |
44
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29 \defineUnit{binomialkoeffizient}{% |
39 | 30 \begin{frame} |
31 \frametitle{Binomialkoeffizient} | |
32 | |
33 \begin{definition}[Binomialkoeffizient] | |
34 Der \structure{Binomialkoeffizient $\binom{n}{k}$} gibt die Anzahl der $k$-elementigen Teilmengen einer $n$-elementigen Menge an. | |
35 \begin{align} | |
36 \binom{n}{k} = \frac{n!}{k!(n-k)!} = \frac{n^{\underline k}}{k!} | |
37 \end{align} | |
38 Man sagt \structure{n über k} oder \structure{k aus n}. | |
39 \end{definition} | |
40 \begin{itemize} | |
41 \item $\binom{n}{k}$ viele Möglichkeiten, $k$ Elemente aus $n$ Elementen zu wählen | |
42 \end{itemize} | |
43 | |
44 \vfill | |
45 | |
44
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46 \begin{theorem}[Pascalsche Identität] |
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47 Die \structure{Pascalsche Identität} liefert eine rekursive Definition des Binomialkoeffizienten. |
41 | 48 \begin{align} |
44
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49 \binom{n}{k} = \binom{n-1}{k} + \binom{n-1}{k-1} |
41 | 50 \end{align} |
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51 \end{theorem} |
39 | 52 \end{frame} |
44
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53 } |
39 | 54 |
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55 \defineUnit{multimengen}{% |
39 | 56 \begin{frame} |
57 \frametitle{Multimengen} | |
58 | |
59 \begin{definition}[Multimenge] | |
60 \structure{Multimengen} sind eine Verallgemeinerung gewöhnlicher Mengen.\\ | |
61 Elemente können nun mehrfach vorkommen, die Reihenfolge spielt weiterhin keine Rolle.\\ | |
62 Sie werden meist auch mit $\left\{ \cdot \right\}$ notiert, alternativ $\{\!\vert \cdot \vert\!\}$. | |
63 \end{definition} | |
64 | |
65 \begin{theorem}[Anzahl von Multiteilmengen] | |
66 Eine \structure{$k$-Multiteilmenge} von $M$ mit $\abs{M} = n$ ist eine Multimenge, die $k$ (nicht unbedingt verschiedene) Elemente aus $M$ enthält.\\ | |
67 Es gibt | |
68 \begin{align} | |
69 \structure{\binom{k + n - 1}{k}} = \binom{k + n - 1}{n - 1} | |
70 \end{align} | |
71 solche Multiteilmengen. | |
72 \end{theorem} | |
73 | |
74 \begin{example}[] | |
75 \begin{itemize} | |
76 \item $M \defeq \left\{ 1, 2, 2, 2, 3 \right\} = \left\{ 2, 1, 2, 3, 2 \right\} \qquad \abs{M} = 5$ | |
77 \end{itemize} | |
78 \end{example} | |
79 \end{frame} | |
80 } | |
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81 } |
39 | 82 |
83 \defineUnit{doppeltesabzaehlen}{% | |
84 \begin{frame} | |
85 \frametitle{Doppeltes Abzählen} | |
86 | |
87 \begin{block}{Doppeltes Abzählen} | |
88 Ermittelt man die \structure{Mächtigkeit} einer Menge auf zwei Arten, so müssen beide Ergebnisse \structure{übereinstimmen}.\\ | |
89 Eine so ermittelte Gleichung kann die gesuchte Mächtigkeit festlegen. | |
90 \end{block} | |
91 | |
92 \begin{example}[Matrizen] | |
43 | 93 In einer Matrix müssen die Summen von Zeilensummen und Spaltensummen übereinstimmen. |
39 | 94 \end{example} |
95 | |
96 \begin{example}[Studenten] | |
97 In einer Vorlesung sitzen \structure{64 Studenten} und \alert{n Studentinnen}.\\ | |
98 Jeder Student kennt genau \structure{5} Studentinnen und jede Studentin \alert{8}~Studenten. | |
99 Wenn \enquote{bekannt sein} symmetrisch ist, wie viele Studentinnen besuchen die Vorlesung? | |
100 { | |
101 \setlength{\belowdisplayskip}{0pt} | |
102 \begin{align} | |
103 \structure{64 \cdot 5} &= \alert{n \cdot 8}\\ | |
104 n &= \frac{64 \cdot 5}{8} = 40 | |
105 \end{align} | |
106 } | |
107 \end{example} | |
108 \end{frame} | |
109 } | |
110 | |
111 \defineUnit{schubfachprinzip}{% | |
112 \begin{frame} | |
113 \frametitle{Schubfachprinzip} | |
114 | |
115 \begin{definition}[Schubfachprinzip] | |
116 Sei $f : X \to Y$ eine Abbildung und $\abs{X} > \abs{Y}$.\\ | |
117 Dann gilt | |
118 \begin{align} | |
119 \exists y \in Y.\, \abs{f^{-1}(y)} \geq \alert{2} | |
120 \end{align} | |
43 | 121 Wenn man \structure{n} Elemente auf \structure{m < n} Fächer verteilt, dann gibt es \structure{mindestens ein Fach}, das mindestens \structure{2} Elemente enthält. |
39 | 122 \end{definition} |
123 | |
124 \vfill | |
125 | |
126 \begin{definition}[Verallgemeinertes Schubfachprinzip] | |
127 Sei $f : X \to Y$ eine Abbildung und $\abs{X} > \abs{Y}$.\\ | |
128 Dann gilt | |
129 \begin{align} | |
130 \exists y \in Y.\, \abs{f^{-1}(y)} \geq \alert{\left \lceil \frac{\abs{X}}{\abs{Y}}\right \rceil} | |
131 \end{align} | |
43 | 132 Wenn man \structure{n} Elemente auf \structure{m < n} Fächer verteilt, dann gibt es \structure{mindestens ein Fach}, das mindestens \structure{$\left\lceil\frac{\abs{X}}{\abs{Y}} \right\rceil$} Elemente enthält. |
39 | 133 \end{definition} |
134 \end{frame} | |
135 } | |
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136 |
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137 \defineUnit{inklusionexklusion}{% |
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138 \begin{frame} |
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139 \frametitle{Inklusion und Exklusion} |
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140 |
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141 \begin{block}{Inklusion und Exklusion} |
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142 Das Prinzip der \structure{Inklusion und Exklusion} erweitert die Summenregel um \alert{nicht disjunkte} Mengen.\\ |
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143 Für drei Mengen $A, B, C$ gilt |
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144 \begin{align} |
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145 \abs{A \cup B \cup C} = \abs{A} &\structure{+} \abs{B} \structure{+} \abs{C}\\ |
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146 &\alert{-} \abs{A \cap B} \alert{-} \abs{A \cap C} \alert{-} \abs{B \cap C}\\ |
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147 &\structure{+} \abs{A \cap B \cap C} |
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148 \end{align} |
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149 \end{block} |
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150 \vfill |
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151 \begin{center} |
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152 \begin{tikzpicture}[x=2em, y=2em] |
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153 \tikzstyle{opa} = [fill opacity=0.5] |
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154 \tikzstyle{A} = [tumred, fill=tumred!35, opa] |
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155 \tikzstyle{B} = [tumblue, fill=tumblue!35, opa] |
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156 \tikzstyle{C} = [tumgreen, fill=tumgreen!35, opa] |
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157 \draw[A] (-2, 0) ellipse (3 and 2); |
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158 \draw[B] (2, 0) ellipse (3 and 2); |
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159 \draw[C] (0, 2) ellipse (3 and 2); |
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160 |
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161 \draw |
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162 (-3.5, -.5) node[tumred] {A} |
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163 (3.5, -.5) node[tumblue] {B} |
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164 (0, 3.5) node[tumgreen] {C} |
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165 (-2, -.5) node {1} |
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166 (2, -.5) node {1} |
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167 (0, 2.5) node {1} |
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168 (-1.5, 1) node {2} |
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169 (1.5, 1) node {2} |
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170 (0, -.5) node {2} |
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171 (0, .75) node {3}; |
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172 \end{tikzpicture} |
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173 \end{center} |
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174 \end{frame} |
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175 } |
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176 |
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177 \defineUnit{stirlingzahlen}{% |
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178 \begin{frame} |
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179 \frametitle{Mengenpartition} |
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180 |
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181 \begin{definition}[$k$-Partition] |
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182 Eine \structure{$k$-Partition} einer Menge $A$ ist eine Zerlegung von $A$ in $k$ \alert{disjunke, nichtleere Teilmengen} $A_1, \dots, A_k$ mit |
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183 \begin{align} |
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184 \biguplus_{i=1}^k A_i = A |
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185 \end{align} |
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186 Dabei bezeichnet $\uplus$ die disjunkte Vereinigung. |
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187 \end{definition} |
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188 |
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189 \vfill |
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190 |
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191 \begin{example}[] |
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192 Einige mögliche \structure{$3$-Partitionen} von $[5]$ sind |
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193 \begin{align} |
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194 \left\{ \left\{ 1,2 \right\}, \left\{ 3,4 \right\}, \left\{ 5 \right\} \right\} &&& |
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195 \left\{ \left\{ 1 \right\}, \left\{ 3,4 \right\}, \left\{ 2, 5 \right\} \right\}\\ |
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196 \left\{ \left\{ 1,2,3 \right\}, \left\{ 4 \right\}, \left\{ 5 \right\} \right\} &&& |
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197 \left\{ \left\{ 1, 5 \right\}, \left\{ 2, 4 \right\}, \left\{ 3 \right\} \right\} |
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198 \end{align} |
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199 Es existieren genau 25 solche $3$-Partitionen. |
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200 \end{example} |
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201 \end{frame} |
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202 |
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203 \begin{frame} |
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204 \frametitle{Stirlingzahlen zweiter Art} |
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205 |
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206 \begin{definition}[Stirlingzahlen zweiter Art] |
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207 Die \structure{Stirlingzahlen zweiter Art $S_{n, k}$} gibt die Anzahl der $k$-Partitoinen einer $n$-elementigen Menge an. |
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208 Wir schreiben |
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209 \begin{align} |
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210 \stirlingtwo{n}{k} &\defeq S_{n, k}\\ |
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211 \intertext{Es ist} |
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212 \stirlingtwo{n}{k} &= \stirlingtwo{n-1}{k-1} + k \cdot \stirlingtwo{n-1}{k} |
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213 \end{align} |
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214 \end{definition} |
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215 \begin{itemize} |
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216 \item $\stirlingtwo{n}{k}$ viele Möglichkeiten, $n$ unterscheidbare Objekte in $k$ gleiche Fächer zu verteilen, sodass jedes Fach ein Objekt bekommt |
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217 \end{itemize} |
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218 |
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219 \vfill |
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220 |
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221 \begin{example}[] |
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222 \begin{itemize} |
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223 \item Es gibt $\stirlingtwo{5}{3} = 25$ $3$-Partitionen von $[5]$. |
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224 \end{itemize} |
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225 \end{example} |
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226 \end{frame} |
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227 |
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228 \begin{frame} |
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229 \frametitle{Permutationen} |
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230 |
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231 \begin{definition}[Permutation] |
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232 Eine \structure{Permutation} einer Menge $A = \left\{ a_1, \dots, a_n \right\}$ ist eine \alert{bijektive Abbildung} $\pi : A \to A$.\\ |
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233 Wir notieren Permutationen in zweizeiligen Vektoren. |
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234 \begin{align} |
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235 \pi = \begin{pmatrix} |
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236 a_1 & \dots & a_n \\ |
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237 \pi(a_1) & \dots & \pi(a_n) |
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238 \end{pmatrix} |
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239 \end{align} |
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240 \end{definition} |
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241 \begin{itemize} |
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242 \item Weist jedem Element in $A$ ein neues, eindeutiges Element in $A$ zu. |
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243 \item \enquote{Mischt} die Elemente einer Menge |
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244 \end{itemize} |
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245 |
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246 \vfill |
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247 |
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248 \begin{example}[] |
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249 $\pi$ ist eine Permutation auf $[9]$. |
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250 \begin{align} |
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251 \pi = \begin{pmatrix} |
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252 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ |
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253 3 & 5 & 4 & 7 & 2 & 6 & 1 & 9 & 8 |
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254 \end{pmatrix} |
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255 \end{align} |
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256 Es ist $\pi(1) = 3$, $\pi(4) = 7$. |
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257 \end{example} |
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258 \end{frame} |
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259 |
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260 \begin{frame} |
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261 \frametitle{Zyklus} |
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262 |
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263 \begin{definition}[$k$-Zyklus] |
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264 Ein \structure{$k$-Zyklus} ist eine Permutation $\pi$, die $k$ verschiedene Zahlen $i_1, \dots, i_k$ im Kreis vertauscht. |
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265 \begin{align} |
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266 \pi &= \begin{pmatrix} |
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267 i_1 & i_2 & \dots & i_k \\ |
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268 i_2 & i_3 & \dots & i_1 |
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269 \end{pmatrix} \\ |
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270 \intertext{Wir schreiben auch} |
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271 \pi &= \begin{pmatrix} |
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272 i_1 & i_2 & \dots & i_k |
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273 \end{pmatrix} |
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274 \end{align} |
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275 Jede Permutation ist eine Verkettung disjunkter Zyklen. |
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276 \end{definition} |
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277 |
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278 \vfill |
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279 |
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280 \begin{example}[] |
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281 % This is ugly :/ |
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282 \vspace{-1em} |
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283 \begin{align} |
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284 \pi &= \begin{pmatrix} |
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285 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ |
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286 3 & 5 & 4 & 7 & 2 & 6 & 1 & 9 & 8 |
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287 \end{pmatrix} \\ |
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288 \intertext{$\pi$ enthält vier Zyklen.} |
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289 \pi &= \begin{pmatrix} |
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290 1 & 3 & 4 & 7 |
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291 \end{pmatrix} \begin{pmatrix} |
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292 2 & 5 |
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293 \end{pmatrix} \begin{pmatrix} |
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294 6 |
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295 \end{pmatrix} \begin{pmatrix} |
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296 8 & 9 |
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297 \end{pmatrix} |
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298 \end{align} |
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299 \end{example} |
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300 \end{frame} |
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301 |
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302 \begin{frame} |
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303 \frametitle{Stirlingzahlen erster Art} |
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304 |
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305 \begin{definition}[Stirlingzahlen erster Art] |
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306 Die \structure{Stirlingzahlen erster Art $s_{n, k}$} gibt die Anzahl der Permutationen mit $n$ Elementen und \alert{k Zyklen} an. |
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307 Wir schreiben |
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308 \begin{align} |
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309 \stirlingone{n}{k} &\defeq s_{n, k}\\ |
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310 \intertext{Es ist} |
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311 \stirlingone{n}{k} &= \stirlingone{n-1}{k-1} + (n-1) \cdot \stirlingone{n-1}{k} |
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312 \end{align} |
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313 \end{definition} |
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314 |
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315 \begin{itemize} |
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316 \item Es gilt $\sum_{k=1}^n \stirlingone{n}{k} = n!$ |
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317 \end{itemize} |
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318 |
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319 \vfill |
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320 |
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321 \begin{example}[] |
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322 \begin{itemize} |
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323 \item Es gibt $\stirlingone{9}{4} = 67284$ Permutationen über $[9]$ mit \alert{vier Zyklen}. |
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324 \end{itemize} |
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325 \end{example} |
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326 \end{frame} |
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327 } |