181 lines
4.4 KiB
TeX
181 lines
4.4 KiB
TeX
\section{Introduction}
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\definition{}
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Let $\Omega$ be an arbitrary set of $n$ objects. \par
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A \textit{permutation} on $\Omega$ is a bijective map $f: \Omega \to \Omega$.
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\vspace{2mm}
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For example, consider the objects 1, 2, and 3. \par
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The permutation $[312]$ is given by a map $f$ defined by the following table:
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\begin{itemize}
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\item $f(1) = 3$
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\item $f(2) = 1$
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\item $f(3) = 2$
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\end{itemize}
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\problem{}
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List all permutations on three objects. \par
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How many permutations of $n$ objects are there?
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\vfill
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\problem{}
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What map corresponds to the permutation $[321]$?
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\vfill
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\problem{}
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What map corresponds to the \say{do-nothing} permutation? \par
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Write it as a function and in square-bracket notation. \par
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\note[Note]{We usually call this the \textit{trivial permutation}}
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\vfill
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\pagebreak
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We can visualize permutations with a \textit{string diagram}, shown below. \par
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The arrows in this diagram denote the image of $f$ for each possible input.
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Two examples are below:
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\vspace{2mm}
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\hfill
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\begin{tikzpicture}[scale=0.5]
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\node (1a) at (0, 0.5) {1};
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\node (2a) at (1, 0.5) {2};
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\node (3a) at (2, 0.5) {3};
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\node (4a) at (3, 0.5) {4};
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\node (1b) at (0, -2) {1};
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\node (3b) at (1, -2) {3};
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\node (4b) at (2, -2) {4};
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\node (2b) at (3, -2) {2};
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\line{1a}{1b}
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\line{2a}{2b}
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\line{3a}{3b}
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\line{4a}{4b}
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\end{tikzpicture}
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\hfill
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\begin{tikzpicture}[scale=0.5]
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\node (1a) at (0, 0.5) {1};
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\node (2a) at (1, 0.5) {2};
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\node (3a) at (2, 0.5) {3};
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\node (4a) at (3, 0.5) {4};
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\node (2b) at (0, -2) {2};
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\node (1b) at (1, -2) {1};
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\node (3b) at (2, -2) {3};
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\node (4b) at (3, -2) {4};
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\line{1a}{1b}
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\line{2a}{2b}
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\line{3a}{3b}
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\line{4a}{4b}
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\end{tikzpicture}
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\hfill\null
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\vspace{2mm}
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Note that in all our examples thus far, the objects in our set have an implicit order.
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This is only for convenience. The elements of $\Omega$ are not ordered (it is a \textit{set}, after all),
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and we may present them however we wish.
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\vspace{1cm}
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For example, consider the diagrams below. \par
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On the left, 1234 are ordered as usual. In the middle, they are ordered alphabetically. \par
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The rightmost diagram uses arbitrary, meaningless labels.
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\vspace{2mm}
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\hfill
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\begin{tikzpicture}[scale=0.5]
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\node (1a) at (0, 0.5) {1};
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\node (2a) at (1, 0.5) {2};
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\node (3a) at (2, 0.5) {3};
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\node (4a) at (3, 0.5) {4};
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\node (2b) at (0, -2) {2};
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\node (1b) at (1, -2) {1};
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\node (3b) at (2, -2) {3};
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\node (4b) at (3, -2) {4};
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\line{1a}{1b}
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\line{2a}{2b}
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\line{3a}{3b}
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\line{4a}{4b}
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\end{tikzpicture}
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\hfill
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\begin{tikzpicture}[scale=0.5]
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\node (4a) at (0, 0.5) {4};
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\node (1a) at (1, 0.5) {1};
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\node (3a) at (2, 0.5) {3};
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\node (2a) at (3, 0.5) {2};
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\node (1b) at (0, -2) {1};
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\node (4b) at (1, -2) {4};
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\node (3b) at (2, -2) {3};
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\node (2b) at (3, -2) {2};
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\line{1a}{1b}
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\line{2a}{2b}
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\line{3a}{3b}
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\line{4a}{4b}
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\end{tikzpicture}
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\hfill
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\begin{tikzpicture}[scale=0.5]
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\node (1a) at (0, 0.5) {$\triangle$};
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\node (2a) at (1, 0.5) {$\divideontimes$};
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\node (3a) at (2, 0.5) {$\circledcirc$};
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\node (4a) at (3, 0.5) {$\boxdot$};
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\node (2b) at (0, -2) {$\divideontimes$};
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\node (1b) at (1, -2) {$\triangle$};
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\node (3b) at (2, -2) {$\circledcirc$};
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\node (4b) at (3, -2) {$\boxdot$};
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\line{1a}{1b}
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\line{2a}{2b}
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\line{3a}{3b}
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\line{4a}{4b}
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\end{tikzpicture}
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\hfill\null
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\vspace{2mm}
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It shouldn't be hard to see that despite the different \say{output} order (2134 and 1432), \par
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the same permutation is depicted in all three diagrams. This example demonstrates two things:
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\begin{itemize}[itemsep=2mm]
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\item First, the names of the items in our set do not have any meaning. \par
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$\Omega$ is just a set of $n$ arbitrary things, which we may label however we like.
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\item Second, permutations are verbs. We do not care about the \say{output} of a certain permutation,
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we care about what it \textit{does}. We could, for example, describe the permutation above as
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\say{swap the first two of four elements.}
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\end{itemize}
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\vspace{2mm}
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Why, then, do we order our elements when we talk about permutations? As noted before, this is for convenience.
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If we assign a natural order to the elements of $\Omega$ (say, 1234), we can identify permutations by simply listing
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their output:
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Clearly, $[1234]$ represents the trivial permutation, $[2134]$ represents \say{swap first two,}
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and $[4123]$ represents \say{cycle right.}
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\problem{}
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Draw string diagrams for $[4123]$ and $[2341]$.
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\vfill
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\pagebreak |