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MathLinks Contest 6th
2.3
0623 bijective mappings 6th edition Round 2 p3
0623 bijective mappings 6th edition Round 2 p3
Source:
May 3, 2021
6th edition
combinatorics
Problem Statement
Let
σ
:
{
1
,
2
,
.
.
.
,
n
}
→
{
1
,
2
,
.
.
.
,
n
}
\sigma : \{1, 2, . . . , n\} \to \{1, 2, . . . , n\}
σ
:
{
1
,
2
,
...
,
n
}
→
{
1
,
2
,
...
,
n
}
be a bijective mapping. Let
S
n
S_n
S
n
be the set of all such mappings and let
d
k
(
σ
)
=
∣
σ
(
k
)
−
σ
(
k
+
1
)
∣
d_k(\sigma) = |\sigma(k) - \sigma(k + 1)|
d
k
(
σ
)
=
∣
σ
(
k
)
−
σ
(
k
+
1
)
∣
, for all
k
∈
{
1
,
2
,
.
.
.
,
n
}
k \in \{1, 2, ..., n\}
k
∈
{
1
,
2
,
...
,
n
}
, where
σ
(
n
+
1
)
=
σ
(
1
)
\sigma (n + 1) = \sigma (1)
σ
(
n
+
1
)
=
σ
(
1
)
. Also let
d
(
σ
)
=
min
{
d
k
(
σ
)
∣
1
≤
k
≤
n
}
d(\sigma) = \min \{d_k(\sigma) | 1 \le k \le n\}
d
(
σ
)
=
min
{
d
k
(
σ
)
∣1
≤
k
≤
n
}
. Find
max
σ
∈
S
n
d
(
σ
)
\max_{\sigma \in S_n} d(\sigma)
max
σ
∈
S
n
d
(
σ
)
.
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