Exact Enumeration of Free Polyplets in the {4,5} Hyperbolic Tessellation

Abstract

Exact Enumeration of Free Polyplets in the {4,5} Hyperbolic Tessellation Status: The exact algebraic extension computed by this repository was officially approved and added to OEIS A390200 on June 3, 2026. Goal: Extend OEIS A390200, the number of free n-celled polyplets in the {4,5} tessellation of the hyperbolic plane. Polyplets are connected by edge or vertex adjacency; each square has 4 edge-neighbors and 8 vertex-neighbors. Sequence Data (A390200): n: 1, 2, 3, 4, 5, 6, 7, 8a(n): 1, 2, 10, 72, 710, 8026, 98353, 1261889 File Structure Polyplets_Exact.py - The primary direct canonical expansion enumerator. Generates polyplets dynamically and canonicalizes their coordinate states against the face stabilizer. Redelmeier-and-Burnside's-Lemma-Verifier.py - An independent enumeration engine utilizing a memory-safe graph spanning tree to generate fixed roots, mathematically computing the free count via the Orbit-Stabilizer theorem. A119611-Verifier.py - A variant of the pipeline configured for strict edge-only connectivity to verify the algebraic core against an established sequence. Coxeter-Verifier.py - A symbolic verification script using SymPy to test that the base matrix generators strictly satisfy the [4,5] Coxeter group relations. Output.json - Structured output data containing candidate counts, computational statistics, execution runtimes, and the verified extension data. Implementation & Methodology The exact backend models cells as cosets of the face stabilizer in the Coxeter group [4,5], using matrices over the algebraic number field Q(2,5). Coefficients are stored exactly as integer numerators over powers of two to completely eliminate floating-point artifacts. The local model rigorously checks: The square stabilizer has order 8 (D4). The vertex stabilizer has order 10 (D5). The polyplet neighbor set has size 12. To guarantee correctness, the repository uses two mathematically distinct enumeration philosophies that corroborate each other: Direct Canonicalization (Polyplets_Exact.py): Expands the boundary dynamically and canonicalizes each finite connected set by translating every cell to the base cell and minimizing over the square stabilizer. Fixed Spanning Tree & Orbit-Stabilizer (Burnside_Enumerator.py): Uses a Redelmeier-style lexicographical spanning tree to generate fixed, rooted animals, bypassing memory constraints. Burnside's Lemma (the Orbit-Stabilizer theorem) is then applied analytically to the symmetries of the exact Coxeter matrices to extract the free count. Verification To prove the algebraic engine, the neighbor generation logic was restricted to edge-only adjacency (s2 reflections) and successfully reproduced the known prefix of OEIS A119611 (Strict {4,5} Polyominoes) perfectly up to n=12. The continuous geometric space is further verified symbolically via Coxeter-Verifier.py. Status & Approved Extension Both the direct canonicalizer and the Burnside enumerator reproduce the full known prefix and mathematically converge on the extended term for n=8. This result has been officially approved and published by the OEIS. n Count Status Candidates Seconds 1 1 ok 1 0.000 2 2 ok 12 0.009 3 10 ok 35 0.029 4 72 ok 238 0.253 5 710 ok 2150 1.951 6 8026 ok 25540 24.498 7 98353 ok 337234 347.390 8 1261889 new 4725668 6753.712 Extended Term:a(8)=1261889 References OEIS A390200 OEIS A119611 arXiv:2109.05331, Extremal p,q-Animals arXiv:2206.14910, Isoperimetric Formulas for Hyperbolic Animals