An Explicit Catalan Face-Factorized Formula for Closed Meander Numbers

We derive an explicit finite nonrecursive formula for the unrestricted closed meander numbers M_n. Starting from the noncrossing-partition permutation model, we express the one-component condition by an exact determinant indicator for an n-cycle and expand this indicator through exterior powers of the natural permutation representation. The resulting matrix coefficients are signed partial-successor counts. Their key feature is a complete planar factorization: for every prescribed noncrossing partial successor map, the number of completions is a product of ordinary Catalan numbers, one factor for each face of the induced partial matching. Thus the summands no longer require the enumeration or connectivity testing of completed meanders. We also derive a local-sign factorization for the noncrossing partial-successor terms and give an independent compact frontier transfer for exact enumeration. Refining the frontier state by the Dyck height at the slit gives an exact ballot-number formula for reachable states and, under the remaining-depth pruning, an exponential state-space growth constant φ² = 2.6180339887..., where φ = (1 + √5)/2. We emphasize that φ² is the growth constant of this frontier-state representation, not the asymptotic growth constant of the meander numbers themselves. A compact implementation reproduces the standard closed-meander numbers through order 22. To the best of our knowledge, the main theorem gives a new explicit finite nonrecursive formula for unrestricted closed meanders in which every completion coefficient reduces to products of ordinary Catalan numbers. We do not claim the first exact representation of meanders, nor a solution of the generating-function or asymptotic meander problem. Version 2.0. Revised and strengthened presentation of the proofs, including a fully explicit treatment of the compact frontier transfer, reachability, and minimal-crossing arguments; expanded literature discussion; and additional clarifications throughout. The main Catalan face-factorized formula is unchanged.