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permutation and expected value

Source: Problem 2, Polish NO 1988

October 16, 2005
symmetryprobabilityexpected valuecombinatorics unsolvedcombinatorics

Problem Statement

For a permutation P=(p1,p2,...,pn)P = (p_1, p_2, ... , p_n) of (1,2,...,n)(1, 2, ... , n) define X(P)X(P) as the number of jj such that pi<pjp_i < p_j for every i<ji < j. What is the expected value of X(P)X(P) if each permutation is equally likely?
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