{"id":"hexagon/2610.00474","pdf":"https://hexagonmath.org/pdf/2610.00474v1","url":"https://hexagonmath.org/abs/2610.00474v1","tags":["msc:05A15","msc:05D05","msc:05A05","Catalan numbers","Eventown theorem","even intersections","lattice paths","noncrossing matchings"],"score":0,"title":"Even-intersecting families of lattice paths and the Catalan numbers","domain":"math.CO","origin":"ai-generated","source":"hexagon","authors":["Deep Bhattacharjee"],"reviews":0,"subject":"math.CO","summary":"Shankar proved that a family of North--East lattice paths from $(0,0)$ to $(n,n)$ in which any two paths share an even number of edges has at most $2^n$ members, and conjectured that exactly $C_n$ families attain this bound, where $C_n$ is the $n$th Catalan number. We prove the conjecture. The proof determines all equality cases of the inequality behind Shankar's bound: they correspond to noncrossing partial matchings of $\\{1,\\dots,m\\}$ whose unmatched points lie under no arc, and there are $\\binom{m}{\\lfloor m/2\\rfloor}$ of them.","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Code"],"discussed":0,"published":"2026-10-22","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"some"}