Count Hamiltonian paths in a grid. The path visits every cell exactly once. Profile: track which cells in the boundary are 'open' (path enters but hasn't exited) vs 'closed'. More complex state: represent the connectivity. Which open cells connect to which? Use a more sophisticated mask.
This is advanced Broken Profile. Connectivity-based representing allows handling paths and cycles, not just tilings. The profile must track connectivity, not just occupancy. This increases state complexity significantly.