Idempotent Rank of the Transformations Semigroup with Palindromic Image

Let $X_n=\{1,2,\ldots,n\}$, and denote by $P_n$ the semigroup of all palindromic transformations on $X_n$. In this paper, we derive an explicit formula for the number of idempotent elements in $P_n$. Furthermore, we prove that $P_n$ is not generated by its idempotents for $n\ge 3$, concluding that its idempotent rank is undefined.