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National and Regional Contests
Mathlinks Contests.
MathLinks Contest 2nd
1.2
1.2
Part of
MathLinks Contest 2nd
Problems
(1)
0212 friendly permutations 2nd edition Round 1 p2
Source:
5/10/2021
We call a permutation
σ
\sigma
σ
of the first
n
n
n
positive integers friendly if and only if the following conditions are fulfilled: (1)
σ
(
k
+
1
)
∈
{
2
σ
(
k
)
,
2
σ
(
k
)
−
1
,
2
σ
(
k
)
−
n
,
2
σ
(
k
)
−
n
−
1
}
,
∀
k
∈
{
1
,
2
,
.
.
.
,
n
−
1
}
\sigma(k + 1) \in \{2\sigma(k), 2\sigma(k) - 1, 2\sigma(k) - n, 2\sigma(k) - n - 1\}, \forall k \in \{1, 2, ..., n - 1\}
σ
(
k
+
1
)
∈
{
2
σ
(
k
)
,
2
σ
(
k
)
−
1
,
2
σ
(
k
)
−
n
,
2
σ
(
k
)
−
n
−
1
}
,
∀
k
∈
{
1
,
2
,
...
,
n
−
1
}
(2)
σ
(
1
)
∈
{
2
σ
(
n
)
,
2
σ
(
n
)
−
1
,
2
σ
(
n
)
−
n
,
2
σ
(
n
)
−
n
−
1
}
\sigma(1) \in \{2 \sigma(n), 2\sigma(n) - 1, 2\sigma(n) - n, 2\sigma(n) - n - 1\}
σ
(
1
)
∈
{
2
σ
(
n
)
,
2
σ
(
n
)
−
1
,
2
σ
(
n
)
−
n
,
2
σ
(
n
)
−
n
−
1
}
. Find all positive integers
n
n
n
for which there exists such a friendly permutation of the first
n
n
n
positive integers.
combinatorics
2nd edition