Permutations #2

17.02.2010 | by Peter

In uno omnia.*

– Athanasius Kircher

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a\in X_n

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\sigma = \left(\sigma\left(1\right),\sigma\left(2\right),\cdots,\sigma\left(n\right)\right)

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(a \; \sigma(a) \; \sigma^2(a) \; \cdots \; \sigma^{|a|-1}(a))

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\sigma = \begin{pmatrix} 1 & 2 & \cdots & n \\ \sigma\left(1\right) & \sigma\left(2\right) & \cdots & \sigma\left(n\right) \end{pmatrix}

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P \cdot \overline{x}=   \begin{pmatrix}     0 & 0 & 0 & 1\\     0 & 0 & 1 & 0\\     0 & 1 & 0 & 0\\     1 & 0 & 0 & 0\\   \end{pmatrix}   \cdot \begin{pmatrix}1 \\ 2  \\ 3 \\ 4 \\\end{pmatrix}   = \begin{pmatrix}4 \\ 3  \\ 2 \\ 1 \\\end{pmatrix}

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 1^{b_1} 2^{b_2} 3^{b_3} \cdots n^{b_n}

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\sigma = \begin{pmatrix} x_1 & x_2 & \cdots & x_n \\ \sigma\left(x_1\right) & \sigma\left(x_2\right) & \cdots & \sigma\left(x_n\right) \end{pmatrix}

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\sigma_2 = \begin{pmatrix} 1 & 2 & 3 & 4 \\ 4 & 3 & 2 & 1 \end{pmatrix}

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\overline{x} =\begin{pmatrix}x_1 \\ x_2 \\ \vdots \\ x_n \\\end{pmatrix}

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P_\sigma \cdot \begin{pmatrix}x_1 \\ x_2 \\ \vdots \\ x_n \\\end{pmatrix} = \begin{pmatrix} \sigma(x_1) \\ \sigma(x_2) \\ \vdots \\ \sigma(x_n) \\\end{pmatrix}

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\sigma = \begin{pmatrix} 1 & 2 & \cdots & n \\ \sigma\left(1\right) & \sigma\left(2\right) & \cdots & \sigma\left(n\right) \end{pmatrix}

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\sigma = \left(\sigma\left(1\right),\sigma\left(2\right),\cdots,\sigma\left(n\right)\right)

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 1^{b_1} 2^{b_2} 3^{b_3} \cdots n^{b_n}

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\sigma_2 = \begin{pmatrix} 1 & 2 & 3 & 4 \\ 4 & 3 & 2 & 1 \end{pmatrix}

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a\in X_n

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Quod…

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… erat…

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… demonstrandum.

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* All in one.

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***

 

Peter Bies © 2010

 

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3 Comments to “Permutations #2”

  1. Jochen Hein wrote on

    fett

  2. Peter wrote on

    Spassibo!

  3. Jochen Hein wrote on

    Echt, and then glix denn next kiss.
    Peter, geküsst!

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