Let n be a positive integer and let S_{n} 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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2012-2 sum with a permutation, 3.6 out of 5 based on 20 ratings

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

S. OumPost authorA 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.

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

Hong Kyu Kim구하는 값이 시그마 옆의 값들을 모두 더한 값인가요?

S. OumPost author시그마 안의 식은 분수들를 곱한 것입니다.

S. OumPost authorYes.

장성우n>1이라는 조건이 있어야 하지 않나요?

S. OumPost authorIf n=1, then there will be no terms left to multiply and so it will become trivial.

So please assume that n>1.

HunminParkWhen can we see the result?