Bipartite noncrossing Frontier states for cluster enumeration on rectangular grids

We study binary colorings of the rectangular grid G_m,n through the evolution of their white and black monochromatic clusters. Extending Mansour’s finite-automaton framework, we encode column-by-column configurations by frontier states whose full active-cluster partition is necessarily noncrossing. To refine this planar restriction, we associate with each frontier partition a block-adjacency graph Q(Π), whose vertices are the frontier blocks and whose edges join blocks containing consecutive frontier positions. We prove that Q(Π) is always connected and that the block coloring induced by every reachable state is a proper two-coloring of Q(Π); hence Q(Π) is bipartite. For m = 4, the bipartite criterion is complete: exactly 13 of the 14 noncrossing partitions satisfy it, each admitting precisely two complementary proper block colorings. Explicit witnesses realize all 26 resulting nonempty states, so the complete reachable automaton has 27 states after adjoining the initial state. Using an exact labeled transition certificate, we derive a reduced rational generating function for the total cluster count S_4,n, an explicit closed formula, a minimal linear recurrence of order five, and the asymptotic cluster density under the uniform coloring measure. Exact computations for 1 ≤ m ≤ 8 support the conjecture that reachability is characterized by the bipartiteness of the block-adjacency graph.