Finite-n Estimate of Dedekind Numbers using Layer-Ratio Monte Carlo
學年 115
學期 1
出版（發表）日期 2026-09-11
作品名稱 Finite-n Estimate of Dedekind Numbers using Layer-Ratio Monte Carlo
作品名稱（其他語言）
著者 Tian-Shun Chen; Hao Feng; Haozhe Wang; Chian-Shu Chen; Kilar Zhang
單位
出版者
著錄名稱、卷期、頁數 Mathematics 14(18), p.3300
摘要 Dedekind’s problem counts monotone Boolean functions, equivalently downsets of a Boolean lattice. We recast this enumeration as a finite layer-ratio reconstruction problem for the Whitney numbers of the ranked ideal lattice. An exact adjacent-layer double count expresses each layer ratio through local averages of the number of addable elements and the number of removable elements. Reversible fixed-layer Markov chains estimate these averages and hence estimate the Dedekind number 𝑀(𝑛) . Backtests at 𝑀(8) and 𝑀(9) calibrate seed-level variability under the fixed protocol and measure the observed Monte Carlo budget scaling. The resulting estimate probes the Whitney-number sequence of the ideal lattice. Although these rows have previously been described empirically as unimodal, the high-precision 𝑛=9 estimate has a shallow two-shoulder feature around the central rank, contrary to that empirical description; the 𝑛=11 and 𝑛=13 center-window estimates show a higher-contrast analogous pattern. The protocol estimate for 𝑀(10) is 𝑀̂(10)=(8.9360±0.0010)×1078 , where the displayed uncertainty is the budget-based forecast scale from the cross-n scaling law under the production budget. It is an empirical variability scale associated with the stated Monte Carlo protocol and production budget, estimated from seed-level variation.
關鍵字 Dedekind numbers; monotone Boolean functions; distributive lattices; Whitney numbers; Markov-chain Monte Carlo
語言 en
ISSN 2227-7390
期刊性質 國外
收錄於 SCI
產學合作
通訊作者 Chian-Shu Chen
審稿制度 否
國別 CHE
公開徵稿
出版型式 ,電子版,紙本
相關連結

機構典藏連結 ( http://tkuir.lib.tku.edu.tw:8080/dspace/handle/987654321/129982 )