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Permutations/n and n+1/Bijection/Exercise
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Show that the assignment
S
n
×
{
1
,
…
,
n
+
1
}
⟶
S
n
+
1
,
(
φ
,
x
)
⟼
φ
~
,
{\displaystyle S_{n}\times \{1,\ldots ,n+1\}\longrightarrow S_{n+1},(\varphi ,x)\longmapsto {\tilde {\varphi }},}
given by
φ
~
(
k
)
=
{
φ
(
k
)
for
k
≤
n
and
φ
(
k
)
<
x
,
φ
(
k
)
+
1
for
k
≤
n
and
φ
(
k
)
≥
x
,
x
for
k
=
n
+
1
,
{\displaystyle {}{\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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