Combinatorics of the paths towards synchronization

The paper’s core combinatorial codings and enumerative claims (Dyck-path bijection for KN, q-Catalan/Carlitz–Riordan recurrence for the length distribution, and the bipartite parallelogram-polyomino correspondence with Narayana counts and area–length relation) are substantively correct. However, the paper contains two evident typos that the model explicitly fixes: (i) the formula mapping an increasing function φ to an edge set (Equation (9) in the paper) is incorrect as printed; used literally, it would make Id encode KN rather than the empty graph, contradicting the surrounding text and figures. The model uses the correct “upper-triangular” edge template, which restores consistency. (ii) The initial condition P0=0 for the Carlitz–Riordan polynomials is a misprint; consistency of the standard recurrence requires P0=1. Aside from these fixes, the model’s solution gives a constructive realization for any φ and a clean area interpretation; for the bipartite case it reproduces the paper’s bijection to parallelogram polyominoes and the area–length counting. Therefore, both are correct in substance, with the model providing clarifications and independent proofs.