Closed Meander Numbers: A Canonical Decomposition, an Explicit Formula, and an Asymptotically Negligible Core
Year: 2026
Type: preprint
Abstract: We give a canonical decomposition of closed plane meanders and an explicit finite arithmetic formula for their numbers M_n. The starting point is a cut crossed by exactly one upper and one lower edge. Closing the two sides gives ordinary smaller meanders of orders a,b with a+b=n+1; a fixed cut therefore contributes exactly M_a M_b possibilities. Selecting the first such cut avoids double counting and separates a reconstructible family. In its complement, the unique maximal internal intervals having one outgoing edge on each side of the line contract to a core. Each contracted interval closes to an ordinary smaller meander, with no auxiliary marking. This yields a previous-order contribution B_n, with M_n = B_n + G_n (n >= 3). We then eliminate the unknown core count G_n. A prescribed edge deletion, a reduced incidence determinant, weighted Catalan recursions, and interval constant-term filters give G_n independently of all meander counts. Substitution into Lagrange inversion produces a formula for M_n depending only on n. A mixed-radix specialization evaluates every constant term by integer arithmetic without generating matchings, meanders, spanning trees, or connectivity states. The integer sizes can be very large. Finally, an insertion argument proves limsup (G_n/M_n)^(1/n) <= 2^(-1/16) < 1, so B_n ~ M_n. Exact coefficient calculations through order 38 are reported using the author's supplied reference values; the large-order residuals in these tables are obtained by subtraction. Preparation of the manuscript and supplementary arithmetic-check code involved assistance from OpenAI Codex/ChatGPT. This is a preprint and has not been peer reviewed. Related earlier preprint by the author: https://doi.org/10.5281/zenodo.21871516
Author: Alepis, Efthimios
Institution: University of Piraeus
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225572340