Denote (still) $n\times n$ [permutation matrices][1] by $\mathfrak{S}_n$. The ordinary transpose preserves this group. Given $P\in\mathfrak{S}_n$, construct the $n\times n$ matrix ${}^tP$ according to the rules: (1) leave the 4 rims unchanged (1st row, 1st column, last row, last column); (2) [transpose][2] its $(n-2)\times(n-2)$-submatrix found by removing the 4 rims. **Example.** $$A=\begin{pmatrix} 1&0&0&0\\0&0&0&1\\0&1&0&0\\0&0&1&0 \end{pmatrix} \rightarrow {}^tA=\begin{pmatrix} 1&0&0&0\\0&0&1&1\\0&0&0&0\\0&0&1&0 \end{pmatrix}.$$ **Example.** $$B=\begin{pmatrix} 1&0&0&0\\0&0&0&1\\0&0&1&0\\0&1&0&0 \end{pmatrix} \rightarrow {}^tB=\begin{pmatrix} 1&0&0&0\\0&0&0&1\\0&0&1&0\\0&1&0&0 \end{pmatrix}.$$ Notice that ${}^tB$ is a permutation while ${}^tA$ is not! >**Question.** What is $a_n:=\#\{^{t}P\in\mathfrak{S}_n:\,\,P\in\mathfrak{S}_n\}$, for $\geq3$? Easy: $a_3=6$. **Remark.** This is a modest case we consider which can easily be generalized to the question of transposing several submatrices. [1]: http://mathworld.wolfram.com/PermutationMatrix.html [2]: https://en.wikipedia.org/wiki/Transpose