It is possible to proof that the number of perfect matching on a set of $2n$ elements is $n!!$, and on the other hand, it is also possible to proof that the number of permutations $\varphi$ of a set of $2n$ elements with the property that every cycle of $\varphi$ has even length is $(n!!)^2$. This implies that there exists a bijection between the aforementioned set and the pair of perfect matching. I wonder if someone has a nice interpretation for this bijection; meaning, how can we prove this constructing the bijection directly. Thanks!
2026-03-25 03:07:18.1774408038
Bijection between perfect matchings permutations with even cycles
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