The rank of the inverse semigroup of partial automorphisms on a finite fence

A fence is a particular partial order on a (finite) set, close to the linear order. In this paper, we calculate the rank of the semigroup ${\mathcal{F}\mathcal{I}}_n$ of all order-preserving partial injections on an $n$-element fence. In particular, we provide a minimal generating set for ${\mathcal{F}\mathcal{I}}_n$. In the present paper, $n$ is odd since this problem for even $n$ was already solved by I. Dimitrova and J. Koppitz.