The following question is motivated by my research.
Let's consider a permutation $\sigma$ of the set $\{1, \ldots, n\}$. We define an element $i \in \{1, \ldots, n\}$ to be locally minimal for $\sigma$ if it satisfies the conditions $i \leq \sigma(i)$ and $i \leq \sigma^{-1}(i)$. I am interested in studying the total number of locally minimal elements.
For any fixed point $i$ where $\sigma(i) = i$, it is considered a locally minimal element for $\sigma$.
The values of this function range between 1 (since $i=1$ is locally minimal) and $n$ (the total number of locally minimal elements is only equal to $n$ when $\sigma$ is the identity permutation). It is worth noting that each cycle in the cycle decomposition of $\sigma$ generates at least one locally minimal element.
My question is whether this function, or a related function, is known and has been previously studied. I attempted to locate this function on the website https://www.findstat.org/, but I was unable to find any relevant information.
