Permutations/n and n+1/Bijection/Exercise

Show that the assignment

$S_{n}\times \{1,\ldots ,n+1\}\longrightarrow S_{n+1},(\varphi ,x)\longmapsto {\tilde {\varphi }},$

given by

${}{\tilde {\varphi }}(k)={\begin{cases}\varphi (k){\text{ for }}k\leq n{\text{ and }}\varphi (k)<x\,,\\\varphi (k)+1{\text{ for }}k\leq n{\text{ and }}\varphi (k)\geq x\,,\\x{\text{ for }}k=n+1\,,\end{cases}}\,$

is well-defined and bijective.

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