Topological characterization of permutation signatures in Berg’s pseudoinverse expansion

Characterize the signature of each permutation appearing in Berg’s formula for the Moore–Penrose pseudoinverse ξ^+ purely in terms of topological invariants of the corresponding generalized connection or generalized linear subdigraph in the digraph D(ξ). Develop a precise rule that maps the combinatorial structure (disjoint paths and cycles induced by submatrix restrictions) to the permutation sign in the determinant expansions used within Berg’s formula.

Background

To extend the combinatorial interpretations from ordinary inverses to the Moore–Penrose pseudoinverse, the paper invokes Berg’s formula, which expresses ξ^+ as a weighted sum of inverses of full-rank submatrices. Each such inverse admits determinant expansions indexed by permutations, which the authors interpret via generalized connections and generalized linear subdigraphs in the associated digraph.

The conjecture seeks a topological rule for the permutation sign based solely on structural invariants of these generalized connections/subdigraphs, thereby unifying combinatorial and topological descriptions of ξ^+ in this framework.