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2012-2 sum with a permutation

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

HunminPark February 17, 2012 at 4:08 pm

What is the meaning of pi(1), pi(2), … , pi(n)???
S. Oum Post authorFebruary 17, 2012 at 4:12 pm

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.
HunminPark February 17, 2012 at 4:15 pm

S.Oum// So pi(n)=[nth number of a permutation pi ∈S_n]???
Hong Kyu Kim February 19, 2012 at 1:37 am

구하는 값이 시그마 옆의 값들을 모두 더한 값인가요?
S. Oum Post authorFebruary 19, 2012 at 9:18 am

시그마 안의 식은 분수들를 곱한 것입니다.
S. Oum Post authorFebruary 19, 2012 at 9:19 am

Yes.
장성우 February 20, 2012 at 9:59 pm

n>1이라는 조건이 있어야 하지 않나요?
S. Oum Post authorFebruary 20, 2012 at 10:36 pm

If n=1, then there will be no terms left to multiply and so it will become trivial.
So please assume that n>1.
HunminPark February 24, 2012 at 12:26 am

When can we see the result?

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