Let \(H\) be a set with \(n\) elements and \(\circ\colon H\times H\to{\mathcal P}(H)\) be a partial hyperoperation on \(H\). If the set \(\{(a,b)\in H^2\mid a\circ b\neq\emptyset\}\) has \(k\) elements, where \(k\in\{0,1,\dots,n^2\}\), then the partial hypergroupoid \((H,\circ)\) is said to be of class \(k\). A natural question which raises up is to find the number of distinct and non-isomorphic partial hypergroups of class \(k\) which can be defined on a set with \(n\) elements. In the paper this problem is completely solved for \(k=2\).