2026/07/22 by Boris Kafidov
Mathematics · #math.CO #msc:05A20 #msc:05C15 #msc:05E05
29 pages, no figures. Ancillary exact-arithmetic Python verifiers included; counterexample computer-assisted
arxiv created 2026/08/01 · arxiv updated 2026/08/04
Let XGh(x;q)=∑μ\vdash ncμ(q)eμ(x) be the chromatic quasisymmetric function of the natural unit interval graph attached to a Hessenberg function h. We establish an infinite class, valid in all orders, for which every nonzero polynomial cμ(q) has a nonnegative, log-concave coefficient sequence with interval support. Namely, this holds whenever h is abelian and its complement-Ferrers partition λ satisfies min\λ1,ℓ(λ)\≤ 3; equivalently, the diagram has at most three rows or at most three columns. Cubic interpolation reduces the rank-three case to a uniform theorem for a difference of two products of four q-integers, proved by positive decomposition, interval methods, and finite-window smoothing. The argument also yields explicit formulas for every supported elementary coefficient in complement-Ferrers rank at most three. We also include a connected 13-vertex natural unit interval graph for which one elementary coefficient is positive, palindromic, and unimodal but not log-concave, thereby recording the failure of the unrestricted conjecture. Thus low complement-Ferrers rank gives a substantial positive regime even though coefficientwise e-log-concavity fails in general.