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) \(\cal{C}_n\) is an \(n\)-ordered set with the partial order \(\prec\) on \([n]\), where the only comparabilities are $$1 \prec 2 \succ 3 \prec 4 \succ \cdots \prec n \succ 1 ~~\mbox{ or }~~ 1 \succ 2 \prec 3 \succ 4 \prec \cdots \succ n \prec 1.$$ 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 \(\cal{C}_n\). We consider the elements, determine a generating set of minimal size and calculate the rank of \(IC_n\).