Research Article · Preprint Mathematics
Local comparison counting for linear extensions and a bounded-range balance refinement
- Primary subject
- Mathematics
- Subject subfields
- Combinatorics and Graph Theory
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 2 · Permanent link
Overview
Abstract
For uniform linear extensions of a finite poset with maximum incomparability degree at most D, we give a local stable-partition estimate and track the incomparabilities removed by conditioning. Every consistent specified order of three marked vertices has probability at least F_D^(-1), where F_D=binom(D+⌊ D/2⌋, ⌊ D/2⌋) binom(2D-⌊ D/2⌋, D). This denominator has order Θ((27/4)^D/D). Substitution into the finite-boundary argument of Aires–Chan–Pak–Panova gives a sharper explicit bounded-range balance estimate. A complementary two-chain construction shows that a triple-order estimate cannot universally use even a constant multiple of the sharp pair probability: its rare order has reciprocal probability asymptotic to 16^k/(4π) at range D=2k. These are quantitative counting refinements and a method limitation.
Keywords partially ordered sets · linear extensions · balance · incomparability degree · local comparison counting
Scope of the work
A conditioned local stable-partition counting estimate and explicit three-vertex order bound yield the stated bounded-range balance refinement, with a two-chain obstruction to replacing triple-order probabilities by a universal multiple of the sharp pair probability.
Reproduction notes
The paper has 7 pages. The editable-source archive contains 7 files. Compile source/main.tex. Standard-library reproducibility/verify.py provides quick checks through five vertices or --full checks through six vertices, with exact stored result comparisons.
The manuscript
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Author disclosures
AI tools assisted proof exploration, literature review, verification-code development and manuscript preparation. Independent AI checks are not external human peer review.
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Cite this edition
Scott Schubr. (2026). Local comparison counting for linear extensions and a bounded-range balance refinement (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/local-comparison-counting-linear-extensions-bounded-range-balance-refinement/v/2
BibTeX
@techreport{luminary_local_comparison_counting_linear_extensions_bounded_range_balance_refinement_v2,
title = {Local comparison counting for linear extensions and a bounded-range balance refinement},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 2},
url = {https://research.luminaryoddities.com/local-comparison-counting-linear-extensions-bounded-range-balance-refinement/v/2}
}Loading discussion…