Assume that $\sigma\in S_n$ has the cycle type $(p,p,.,p,1,..,1)$ where $p>2$ is a prime and the numbers of $1$ maybe $0$. If $\sigma_1$ and $\sigma_2$ are chosen uniformly in the conjugacy class of $\sigma$. Assume the cycle type of $\sigma_1 \sigma_2=(k_1,k_2,.,k_l)$ . Is there a lower bound $B$ such that $$\mathrm{Pr}\big(\exists j\in \{1,2,..,l\}, \text{ s.t. } k_j \bmod 2=0\big)\geq B \geq 1/T(n),$$ where $T$ is a polynomial.

Assume that $\sigma\in S_n$ has the cycle type $(p,.,p,1,..,1)$ where $p>2$ is a prime and the numbers of $1$ maybe $0$. If $\sigma_1$ and $\sigma_2$ are chosen uniformly in the conjugacy class of $\sigma$. Assume the cycle type of $\sigma_1 \sigma_2=(k_1,k_2,.,k_l)$ . Is there a lower bound $B$ such that $$\mathrm{Pr}\big(\exists j\in \{1,2,..,l\}, \text{ s.t. } k_j \bmod 2=0\big)\geq B \geq 1/T(n),$$ where $T$ is a polynomial.

Assume that $\sigma\in S_n$ has the cycle type $(p,p,.,p,1,..,1)$ where $p$ is a prime and the numbers of $1$ maybe $0$. If $\sigma_1$ and $\sigma_2$ are chosen uniformly in the conjugacy class of $\sigma$. Assume the cycle type of $\sigma_1 \sigma_2=(k_1,k_2,.,k_l)$ . Is there a lower bound $B$ such that $$\mathrm{Pr}\big(\exists j\in \{1,2,..,l\}, \text{ s.t. } k_j \bmod 2=0\big)\geq B \geq 1/T(n),$$ where $T$ is a polynomial.

Assume that $\sigma\in S_n$ has the cycle type $(p,p,.,p,1,..,1)$ where $p>2$ is a prime and the numbers of $1$ maybe $0$. If $\sigma_1$ and $\sigma_2$ are chosen uniformly in the conjugacy class of $\sigma$. Assume the cycle type of $\sigma_1 \sigma_2=(k_1,k_2,.,k_l)$ . Is there a lower bound $B$ such that $$\mathrm{Pr}\big(\exists j\in \{1,2,..,l\}, \text{ s.t. } k_j \bmod 2=0\big)\geq B \geq 1/T(n),$$ where $T$ is a polynomial.

Assume that $\sigma\in S_n$ has the cycle type $(p,p,.,p,1,..,1)$ where $p$ is an odd prime and the numbers of $1$ maybe $0$. If $\sigma_1$ and $\sigma_2$ are chosen uniformly in the conjugacy class of $\sigma$. Assume the cycle type of $\sigma_1 \sigma_2=(k_1,k_2,.,k_l)$ . Is there a lower bound $B$ such that $Pr(\exists j\in \{1,2,..,l\}, \,s.t. \,k_j \, mod \, 2=0)\geq B \geq 1/T(n)$, where T is a polynomial.

Assume that $\sigma\in S_n$ has the cycle type $(p,p,.,p,1,..,1)$ where $p$ is a prime and the numbers of $1$ maybe $0$. If $\sigma_1$ and $\sigma_2$ are chosen uniformly in the conjugacy class of $\sigma$. Assume the cycle type of $\sigma_1 \sigma_2=(k_1,k_2,.,k_l)$ . Is there a lower bound $B$ such that $$\mathrm{Pr}\big(\exists j\in \{1,2,..,l\}, \text{ s.t. } k_j \bmod 2=0\big)\geq B \geq 1/T(n),$$ where $T$ is a polynomial.