2026/07/27 by Anish Gupta
Mathematics · #math.CO #msc:05A16 #msc:06A07 #msc:68R05
16 pages, 3 figures. Version 2 adds an exhaustive order-14 balance and linear-extension-majority census, including extremal balance constants, the equality locus, and cycle spectra; the Gold Partition result is unchanged. Code, data, and archived computational artifacts: https://doi.org/10.5281/zenodo.21696940
arxiv created 2026/07/30 · arxiv updated 2026/07/31
We determine the exact extremal balance data of all 1,338,193,159,771 unlabeled posets on fourteen elements. The least balance constant exceeding 1/3 is 37/106. The least over posets that are not nontrivial ordinal sums is 254/725, attained by a ladder with broken rungs; this confirms a conjecture of Peczarski at order 14, while orders 12 and 13 reproduce De Loof, De Baets, and De Meyer. No balance constant lies in the gap above 1/3 that Peczarski conjectures to be empty. Exactly 128 classes attain 1/3, and every one is an ordinal sum of singletons and copies of the three-element poset with one relation, a family whose non-chain members are counted by a(n)-1, where a(n)=a(n-1)+a(n-3). In the linear-extension-majority digraph the longest simple cycle has length 8, against 7 at order 13, and exactly 30 classes attain it; of the thirteen such classes whose witnesses the census retains, nine have a cycle spectrum containing no odd cycle at all. A second exhaustive pass over the same classes verifies Peczarski's Gold Partition Conjecture through fourteen elements, extending his order-11 frontier and implying in particular that the 1/3-2/3 Conjecture holds through order 14. All arithmetic is exact and every extremal witness is recomputed by an independent program.