Devise a backtracking algorithm to allocate $p$ projects to $s$ students

Question

There are $p$ projects and $s$ students where $p \le s$. Multiple students can work on the same project but the same student cannot work on multiple projects. All projects must be allocated and all students must have a project. I need to compute and display all possible ways of allocating projects to students under these constraints. Here's an example:

Input:

3 students, 2 projects

Output:

   P1      P2
   {1}     {2, 3}
   {2}     {1, 3}
   {3}     {1, 2}
   {1, 2}  {3}
   {1, 3}  {2}
   {2, 3}  {1}

In my attempts, I've thought about this as being related to the problem of partitioning the set of students into non empty subsets and then ignoring those partitions which have unacceptable length. i.e. for my example generate:

{{1}, {2}, {3}}
{{1}, {2, 3}}
{{2}, {1, 3}}
{{3}, {1, 2}}
{{1, 2, 3}}

and ignore the first and last as well as double count the three in the middle as they can be arranged in a two ways (order matters for this problem). I'm not sure however how I would go about generating and filtering these subsets and even whether this is a decent solution for any p or s values.