Hi, I need to work out the number of all permutations, [itex]\tau[/itex], in the form:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\tau = (\sigma_1 \sigma_2 \ldots \sigma_n) (\theta_1 \theta_2 \ldots \theta_n) \quad \text{for} \quad \sigma_i \neq \theta_j \quad \text{and} \quad \tau \in S_{2n}[/tex]

Namely 2 disjoint cycles of equal length in the symmetric group of degree 2n, letting [itex]\text{o}(\tau)[/itex] be the number of permutations of this form existing, then I have a good guess that:

[tex] \text{o}(\tau) = \frac{(2n)!}{2n^2}[/tex]

My first thought was to try and prove this inductively, but I’m struggling to come up with some kind of sum for [itex]\text{o}(\tau)[/itex]. Could anyone give me a hint or a starting step please.

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# Number of all permutations

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