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Average value of function taken over all permutations

Source: Middle European Mathematical Olympiad 2012 - Team Compt. T-4

September 14, 2012
functionmodular arithmeticnumber theorycombinatorics proposedcombinatoricspermutation statistics

Problem Statement

Let p>2 p>2 be a prime number. For any permutation π=(π(1),π(2),⋯ ,π(p)) \pi = ( \pi(1) , \pi(2) , \cdots , \pi(p) ) of the set S={1,2,⋯ ,p} S = \{ 1, 2, \cdots , p \} , let f(π) f( \pi ) denote the number of multiples of p p among the following p p numbers: π(1),π(1)+π(2),⋯ ,π(1)+π(2)+⋯+π(p) \pi(1) , \pi(1) + \pi(2) , \cdots , \pi(1) + \pi(2) + \cdots + \pi(p) Determine the average value of f(π) f( \pi) taken over all permutations π \pi of S S .
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