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The bijection is just to swap the people sitting in the first and last seats.

I feel like a ‘formula’ is more machinery than this simple concept is worth, but here, if this helps: let $P$ be the set of permutations on $\\{0\ldots n - 1\\}$, where for $p \in P$, $p(s)$ is the number of the passenger who sits in seat $s$. Then there exists a bijection $-\_{\operatorname{swap}} : P \to P$, defined with the following formula:
$$
p\_{\operatorname{swap}}(s) = \left\\{\begin{array}{ll}
\\, p(n - 1) \quad & \text{if} \\, s = 0\\\\
\\, p(s) \quad & \text{if} \\, 0 < s < n - 1\\\\
\\, p(0) \quad & \text{if} \\, s = n - 1
\end{array}\right.
$$

I'm not going to draw a picture of the people in the first and last seats switching places. Please try to imagine that yourself.