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A permutation $\sigma$ is a bijection from the set $[n]=\{1,2,…,n\}$ to itself. We denote it using the notation

 

$$\begin{pmatrix}1 & 2 & … & n \\ {\sigma(1)} & {\sigma(2)} & … & {\sigma(n)} \end{pmatrix},$$

e.g. if n=3 then $\begin{pmatrix}1&2&3 \\ 2&3&1 \end{pmatrix}$ denotes the permutation defined by $\sigma(1)=2, \sigma(2)=3 and \sigma(3)=1.$

An inversion in $\sigma$ is a pair $(i,j)$ such that $i<j$ but $\sigma(i)>\sigma(j).$ The sign of a permutation $\sigma$ (denoted $sgn(\sigma)$) is defined to be $(-1)^{inv(\sigma)}$, where $inv (\sigma)$ denotes the total number of inversions in $\sigma .$ In the above example, there are 2 inversions corresponding to the pairs $(1,3)$ and $(2,3)$ so that $sgn(\sigma)=(-1)^2-1.$ For each permutation $\sigma$, define a matrix $A_\sigma$ as follows:

$$A_{\sigma}(i,j) = \left\{ \begin{array}{rcl} 1 & \text{if } \sigma(i)=j \\ 0 & \text{otherwise } \end{array}\right.$$

What are the determinants $|A_{\sigma} |$ and $|A_\tau|?$ Can you relate these with the signs of permutations $\sigma$ and $\tau ?$
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soujanyareddy13 asked Jan 29, 2021
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A permutation $\sigma$ is a bijection from the set $[n]=\{1,2,\dots ,n\}$ to itself. We denote it using the notation$$\begin{pmatrix}1 & 2 & … & n \\ {\sigma(1)} & {\si...