Are there some references about the following result in the literature of combinatorics?
Let $P$ be a permutation on $\{1,2,\ldots,n\}$. Let $\min P$ be the minimal number in the codomain of $P$. For example, $\min ((135)) = 1$ and $\min ((234)) = 2$. Let $\max P$ be the maximal number in the codomain of a permutation $P$. For example, $\max ((135)) = 5$ and $\max ((234)) = 4$.
We say two cycles $P_1, P_2$ do not cross each other if $\max P_1 < \min P_2$ or $\max P_2 < \min P_1$. For example, $(12)$ and $(35)$ do not cross each other; $(256)$ and $(134)$ cross each other.
I think that the following result is true.
Let $\pi, \pi_1, \pi_2$ be permutations on $\{1,2,\ldots,n\}$ such that $\pi_2 = \pi \pi_1$. Let $\pi=P_1 \cdots P_k$ be a cycle decomposition of $\pi$. Suppose that $k \geq 3$, $\pi_1, \pi_2$ are 321-avoiding, and $P_i, P_j$ cross each other for all $i \neq j, i,j\in\{1,2,\ldots,k\}$. Then $\pi_1$ is the identity permutation or $\pi_1, \pi_2$ are not comparable under the Bruhat order.
Is this result studied in the literature of combinatorics? Thank you very much.
Edit: the codomain of a permutation $P$ is the set of image of $P$. For example, when $P=(125)$, then the codomain of $P$ is $\{1,2,5\}$.
