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Romania Team Selection Test
2014 Romania Team Selection Test
3
Permutation sum
Permutation sum
Source: Romania, 4th TST 2014, Problem 3
December 6, 2014
algebra unsolved
algebra
combinatorics
Problem Statement
Let
n
∈
N
n \in \mathbb{N}
n
∈
N
and
S
n
S_{n}
S
n
the set of all permutations of
{
1
,
2
,
3
,
.
.
.
,
n
}
\{1,2,3,...,n\}
{
1
,
2
,
3
,
...
,
n
}
. For every permutation
σ
∈
S
n
\sigma \in S_{n}
σ
∈
S
n
denote
I
(
σ
)
:
=
{
i
:
σ
(
i
)
≤
i
}
I(\sigma) := \{ i: \sigma (i) \le i \}
I
(
σ
)
:=
{
i
:
σ
(
i
)
≤
i
}
. Compute the sum
∑
σ
∈
S
n
1
∣
I
(
σ
)
∣
∑
i
∈
I
(
σ
)
(
i
+
σ
(
i
)
)
\sum_ {\sigma \in S_{n}} \frac{1}{|I(\sigma )|} \sum_ {i \in I(\sigma)} (i+ \sigma(i))
∑
σ
∈
S
n
∣
I
(
σ
)
∣
1
∑
i
∈
I
(
σ
)
(
i
+
σ
(
i
))
.
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