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Permutation of a Set

Source: 1990 Austrian-Polish Math Competition)

April 4, 2014
functioncombinatorics unsolvedcombinatorics

Problem Statement

Let n>1n>1 be an integer and let f1f_1, f2f_2, ..., fn!f_{n!} be the n!n! permutations of 11, 22, ..., nn. (Each fif_i is a bijective function from {1,2,...,n}\{1,2,...,n\} to itself.) For each permutation fif_i, let us define S(fi)=∑k=1n∣fi(k)−k∣S(f_i)=\sum^n_{k=1} |f_i(k)-k|. Find 1n!∑i=1n!S(fi)\frac{1}{n!} \sum^{n!}_{i=1} S(f_i).
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