  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 $\mathcal{FI}_{n}$
of all order-preserving partial injections on an $n$-element fence. In
particular, we provide a minimal generating set for $\mathcal{FI}_{n}$. In the
present paper, $n$ is odd since this problem for even $n$ was already solved by
I. Dimitrova and J. Koppitz.
