# 2012-2 sum with a permutation

Let n be a positive integer and let Sn be the set of all permutations on {1,2,…,n}. Assume $$x_1+x_2 +\cdots +x_n =0$$ and $$\sum_{i\in A} x_i\neq 0$$ for all nonempty proper subsets A of {1,2,…,n}. Find all possible values of$\sum_{\pi \in S_n } \frac{1}{x_{\pi(1)}} \frac{1}{x_{\pi(1)}+x_{\pi(2)}}\cdots \frac{1}{x_{\pi(1)}+\cdots+ x_{\pi(n-1)}}.$

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## 9 thoughts on “2012-2 sum with a permutation”

1. HunminPark

What is the meaning of pi(1), pi(2), … , pi(n)???

2. S. Oum Post author

A permutation in S_n is a bijective function from {1,2,3,…,n} to {1,2,3,…,n}. Pi(i) is the function value at i.

3. HunminPark

S.Oum// So pi(n)=[nth number of a permutation pi ∈S_n]???

4. Hong Kyu Kim

구하는 값이 시그마 옆의 값들을 모두 더한 값인가요?

5. S. Oum Post author

시그마 안의 식은 분수들를 곱한 것입니다.

6. S. Oum Post author

Yes.

7. 장성우

n>1이라는 조건이 있어야 하지 않나요?

8. S. Oum Post author

If n=1, then there will be no terms left to multiply and so it will become trivial.
9. HunminPark