I am looking for a bijection between permutations in $\mathfrak S_n$ with a certain weight and a second set, which arises by interpreting the expression $$ \frac{1}{2}\left(1 + \exp(q \log\left(\frac{1+x}{1-x}\right)\right) $$ as the exponential generating function for permutations with only odd cycles, where each cycle is coloured blue or red, up to interchanging the colours. Each cycle carries the weight $q$. For example, there are $4!=24$ cycles of length 5, $\binom{5}{2}\cdot 2\cdot 2^2 = 80$ coloured permutations with a 3-cycle and two singletons, and $2^4 = 16$ coloured permutations with 5 singletons.

On the other hand, the weight of a permutation in $\mathfrak S_n$ is obtained by computing https://www.findstat.org/StatisticsDatabase/St000389oMp00093oMp00127oMp00066oMp00090 : write down the cycle decomposition of the permutation, cycles ordered by minimal elements, and such that the minimal element of each cycle comes first. Then erase the parenthesis and interpret the result as a permutation $\sigma$ in one-line notation. Finally, interpret this permutation as a Dyck path - drawn as a subdiagonal path from $(0,0)$ to $(n,n)$ - with peaks at $\{(i, \sigma_i-1) | \sigma_i\text{ is a right-to-left minimum of }\sigma\}$. For example, the permutation $2,4,3,1$ has cycle decomposition $(1,2,4)(3)$, thus $\sigma=1,2,4,3$, so the Dyck path has peaks at $(4,3-1)$, $(2,2-1)$, and $(1, 1-1)$. Each ascent of odd length of this Dyck path carries the weight $q$.

This question arises by comparing this answer with this, and the following comments.