Let $\mathrm{ODD}(n)$ be the set of permutations in $\mathfrak{S}_n$ whose cycle lengths are all odd. It is known that $$ \#\mathrm{ODD}(n) = \begin{cases} ((n-1)!!)^2 &\textrm{ if $n$ is even}; \\ n\cdot((n-2)!!)^2 &\textrm{ if $n$ is odd}. \end{cases}$$

Set $\mathcal{C}_{\mathrm{ODD}}(n,t) = \sum_{\pi \in \mathrm{ODD}(n)} t^{\kappa(\pi)}$, where $\kappa(\pi)$ is the number of cycles of $\pi \in \mathfrak{S}_n$.

Then it is easy to show that $$ \sum_{n\geq0}\mathcal{C}_{\mathrm{ODD}}(n,t) \cdot \frac{z^n}{n!} = \left(\frac{\sqrt{1-z^2}}{1-z}\right)^t$$ From this one can show that $$ \mathcal{C}_{\mathrm{ODD}}(n,t)= \sum_{i=0}^{\lfloor n/2\rfloor}\frac{1}{2^i\cdot i!}\prod_{j=0}^{n-1-2i}(t+j)\prod_{k=0}^{i-1}(t-2k)(n-2k)(n-2k-1)$$ In particular if we set $t:=2k$ to be an even integer, then $$ \mathcal{C}_{\mathrm{ODD}}(n,2k)=\frac{(n-1)!}{(2k-1)!} \cdot \sum_{j=0}^{k}(-1)^j\binom{k}{j}\prod_{i=0}^{k-j-1}(n+2i)(n+2i+1)\prod_{i=0}^{j-1}(n-2i)(n-2i-1)$$ Thus, for example $\mathcal{C}_{\mathrm{ODD}}(n,2)=2\cdot n!$ and $\mathcal{C}_{\mathrm{ODD}}(n,4)=4n\cdot n!$.

Define the polynomial $P_k(x)$ by setting $$ \mathcal{C}_{\mathrm{ODD}}(n,2k) = \frac{(n-1)!\cdot k! \cdot 2^k}{(2k-1)!}\cdot P_k(n)$$

For example,

- $P_1(x)=x$;
- $P_2(x)=3x^2$;
- $P_3(x)=10x^3+5x$;
- $P_4(x)=35x^4+70x^2$;
- $P_5(x)=7\cdot 3^2(2x^5+10x^3+3x)$;
- $P_6(x)=3\cdot 7\cdot 11(2x^6+20x^4+23x^2)$;
- et cetera

**Conjecture 1**: $P_k(x)$ is a polynomial of degree $k$ with nonnegative integer coefficients, with zero constant term, and which is odd if $k$ is odd and even if $k$ is even.

**Conjecture 2**: We have,
$$ \sum_{k \geq 0} \frac{P_k(x)}{(2k-1)!!} \cdot z^k = \frac{1}{2}\cdot\left( \frac{1+z}{1-z}\right)^x$$
(The constant term on the RHS is $1/2$ so take whatever convention for $P_0(x)$ or $(-1)!!$ for that to work.)

With regard to Conjecture 2, note that from the above we have $$ \sum_{n\geq0}\mathcal{C}_{\mathrm{ODD}}(n,2k) \cdot \frac{z^n}{n!} = \left(\frac{1+z}{1-z}\right)^k$$

**Question**: Are these conjectures correct? Are these cycle generating functions studied somewhere?

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