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enter image description hereI've tried to compute the length of (321) and I got 2. Then the sgn should be (-1)^2=1. But I suppose sgn(321)=-1 by the definition in the graph?

Del Rey
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    The length of $(321)$ is $3$. The number of transpositions when $(321)$ is written as the product of transpositions is even. – amWhy Mar 10 '15 at 16:03
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    The signature of a cycle of length $k$ is $(-1)^{k-1}$. – Bernard Mar 10 '15 at 16:20

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