Say $S\_n$ is the symmetric group. Define a graph $G$ by $G=(S\_n,E)$, where there is an edge from $\sigma\_1$ to $\sigma\_2$ if and only if $\sigma\_2=t\sigma\_1$ for some transposition $t$.

**Is** there a Hamiltonian path in $G$? If yes, is there a convenient way to define such a path?

No work so far - sue me. Context: For no practical reason, thinking about an elegant implementation of the formula $$\det(A)=\sum\_{\sigma\in S\_n}\text{sgn}(\sigma)\prod\_{j=1}^n a\_{j,\sigma(j)}.$$