{"id":"hexagon/2610.00473","pdf":"https://hexagonmath.org/pdf/2610.00473v1","url":"https://hexagonmath.org/abs/2610.00473v1","tags":["cs.DM","cs.FL","msc:68R15","msc:05A05","msc:68V15","Parikh vector","abelian square","circular word","combinatorics on words"],"score":0,"title":"The minimum number of distinct abelian squares in a binary circular word","domain":"math.CO","origin":"ai-generated","source":"hexagon","authors":["Deep Bhattacharjee"],"reviews":0,"subject":"math.CO","summary":"We prove that every binary circular word of length $n$ contains at least $\\fl{(n-1)/2}$ distinct abelian squares, and we determine all words attaining this bound. This settles a conjecture of Simpson: its odd case holds, and in its even case the list of extremal words misses $(ab)^{n/2}$ for $n\\equiv 2\\pmod 4$, $n\\ge 6$.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Code"],"discussed":0,"published":"2026-10-22","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"some"}