The rank of the inverse semigroup of all partial automorphisms on a finite crown

For $$n \in \mathbb N$$ , let $$[n] = \{1, 2, \ldots , n\}$$ be an n - element set. As usual, we denote by $$I_n$$ the symmetric inverse semigroup on [n], i.e. the partial one-to-one transformation semigroup on [n] under composition of mappings. The crown (cycle) $$\mathcal{C}_n$$ is an n-ordered set with the partial order $$\prec $$ on [n], where the only comparabilities are $$\begin{aligned} 1 \prec 2 \succ 3 \prec 4 \succ \cdots \prec n \succ 1 ~~ \text{ or } ~~ 1 \succ 2 \prec 3 \succ 4 \prec \cdots \succ n \prec 1. \end{aligned}$$ We say that a transformation $$\alpha \in I_n$$ is order-preserving if $$x \prec y$$ implies that $$x\alpha \prec y\alpha $$ , for all x, y from the domain of $$\alpha $$ . In this paper, we study the inverse semigroup $$IC_n$$ of all partial automorphisms on a finite crown $$\mathcal{C}_n$$ . We consider the elements, determine a generating set of minimal size and calculate the rank of $$IC_n$$ .