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I would like to define the following sign for a given perfect matching $P$ of set of $2n$ elements:

$$\sigma_P=(-1)^k$$

where $k$ is the number of crossings in the chord diagram associated to $P$.

Is there a (well known) way to read off $k$ or $k\pmod 2$ directly from the matching $P$, without drawing the diagram itself?

E.g., given the 6-element matching $(13)(24)(56)$ can I tell that the number of crossings in its chord diagram is $k=1$ and hence that the sign is $\sigma=-1$?

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