{"id":"hexagon/2610.00729","pdf":"https://hexagonmath.org/pdf/2610.00729v1","url":"https://hexagonmath.org/abs/2610.00729v1","tags":["math.RA","math.RT","math.FA","msc:15A15","msc:05E05","msc:15B48","msc:20C30","msc:32M15","MacMahon master theorem","Macdonald permanent"],"score":0,"title":"The Macdonald $(q,t)$-permanent I: Algebra and positivity","domain":"math.CO","origin":"ai-generated","source":"hexagon","authors":["Suvrit Sra"],"reviews":0,"subject":"math.CO","summary":"We introduce the Macdonald $(q,t)$-permanent, a two-parameter cycle-weighted permanent that contains the usual permanent and determinant, as well as the $\\alpha$-permanent of Vere-Jones, as degenerate cases. We prove a MacMahon-type master theorem for its generating function, and determine when it induces a valid kernel on tuples of vectors. Nonnegativity of the $(q,t)$-permanent itself on semidefinite matrices turns out to be subtler: for $t=q^\\alpha$ nonnegativity holds exactly at the positive Wallach exponents; for a fixed $q$ we determine the region in rank two and for higher ranks under…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["GPT Pro","Claude Opus 5.5 and Fable 5.1"],"discussed":0,"published":"2026-11-02","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"some"}