Miss Green is very fat and she is very heavy. So she goes to see her doctor. The doctor says, “There isn't anything wrong with you, but you eat too much. I think you must go on a diet. Here's a book for you to read. You must eat the things on page 15. Read it carefully and eat the things for ten days, and then come to me again.”

    Miss Green takes the book home and reads it carefully, and begins her diet.

    Ten days is over now. Miss Green comes to see her doctor again. The doctor finds she is still very fat. So he says, “Do you eat the things on page 15 of that small book now?” “Yes, I do.” Miss Green says. The doctors cannot say anything.

    The next day, the doctor goes to see Miss Green in the afternoon. It is teatime. Miss Green is eating potatoes; “Why are you eating potatoes? They're not in your diet.”

    “But doctor,” Miss Green says, “This is teatime. I eat my diet on breakfast time, lunchtime and suppertime. This diet doesn't tell me anything about my tea.”

设a₁，a₂，…，a_n为1，2，…，n按任意顺序做成的一个排列，f_k是集合{a_i|a_i＜a_k，i＞k}元素的个数，而g_k是集合{a_i|a_i＞a_k，i＜k}元素的个数（k=1，2，…，n），规定f_n=g₁=0，例如：对于排列3，1，2，f₁=2，f₂=0，f₃=0

（I）对于排列4，2，5，1，3，求$\sum _{k=1}^n$$f_k$

（II）对于项数为2n﹣1 的一个排列，若要求2n﹣1为该排列的中间项，试求$\sum _{k=1}^n$$g_k$的最大值，并写出相应得一个排列

（Ⅲ）证明$\sum _{k=1}^n$$f_k$=$\sum _{k=1}^n$$g_k$

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