Enumeration of maximal elements for two-loop separating varieties

Derive a formula for the number of maximal elements of each rank in the poset P_{p,2}, equivalently determining the number of irreducible components of each rank in the separating variety for simultaneous conjugation of two p×p matrices.

Background

For n=2, maximality in the poset P_{p,2} depends on both the ordered partition π and the permutation σ, unlike the simpler n≥3 case where maximality depends only on whether σ is a partial reversal.

The paper gives enough information to determine the subdimension of the separating variety, but not a general enumeration of maximal elements by rank.