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Romania TST 5 2017 P2

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February 25, 2021
combinatoricspermutationsRomanian TST2017

Problem Statement

Let nn be a positive integer, and let SnS_n be the set of all permutations of 1,2,...,n1,2,...,n. let kk be a non-negative integer, let an,ka_{n,k} be the number of even permutations σ\sigma in SnS_n such that ∑i=1n∣σ(i)−i∣=2k\sum_{i=1}^{n}|\sigma(i)-i|=2k and bn,kb_{n,k} be the number of odd permutations σ\sigma in SnS_n such that ∑i=1n∣σ(i)−i∣=2k\sum_{i=1}^{n}|\sigma(i)-i|=2k. Evaluate an,k−bn,ka_{n,k}-b_{n,k}.
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