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Putnam 2005 B6

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December 5, 2005
Putnamintegrationlinear algebramatrixalgebrapolynomialcalculus

Problem Statement

Let SnS_n denote the set of all permutations of the numbers 1,2,…,n.1,2,\dots,n. For π∈Sn,\pi\in S_n, let σ(π)=1\sigma(\pi)=1 if π\pi is an even permutation and σ(π)=−1\sigma(\pi)=-1 if π\pi is an odd permutation. Also, let v(π)v(\pi) denote the number of fixed points of π.\pi. Show that ∑π∈Snσ(π)v(π)+1=(−1)n+1nn+1. \sum_{\pi\in S_n}\frac{\sigma(\pi)}{v(\pi)+1}=(-1)^{n+1}\frac{n}{n+1}.
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