There has to be a way to use rook polynomials to solve this question:

How do I find the number of ways 8 people can be formed into pairs with the constraint that they cannot work with the same person they worked with [on a previous] project?

The restricted cells on the board would naturally be the diagonal (participants cannot be paired with themselves), as well as the cells corresponding to the pairings in the first project.

There are $105$ possible pairs of $8$ people.

In addition, the pairs of participants force symmetry along the diagonal.

The calculus of the coefficients is probably in the sequence AO54479 - OEIS as noted in one of the answers. However, I would like to see how these coefficients can be built up from the ground up.