关于色拟对称函数的\(e -\)对数凹性的一个\(13\)顶点反例
Log-concavity of elementary coefficients for low-rank abelian Hessenberg graphs, with a counterexample in general
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中文总结 AI 辅助
研究色拟对称函数的\(e -\)对数凹性，通过展示一个\(13\)顶点连通自然单位区间图，给出其色拟对称函数的相关系数，反驳了Abreu - Nigro猜想5.3等，计算经独立确定性验证器用精确运算认证。
中文摘要 AI 辅助
我们展示了一个具有\(13\)个顶点的连通自然单位区间图，其色拟对称函数的一个基本基系数是正的、回文的且单峰的，但不是对数凹的。对于黑森伯格函数\(h=(2,4,4,6,7,10,10,10,10,12,12,13,13)\)和\(\lambda=(6,5,1,1)\)，在\([e_{\lambda}]X_{G(h)}(\mathbf{x};q)\)中\(q^5,q^6,q^7\)的系数分别为\(1,6,38\)，即\(6^2<1\cdot38\)。这反驳了Abreu - Nigro的猜想5.3及其后来由Tom和Sagan - Tom给出的表述。计算由一个独立的确定性验证器使用精确整数和有理数运算进行了验证。
英文摘要
Let $X_{G_h}(\mathbf{x};q)=\sum_{μ\vdash n}c_μ(q)e_μ(\mathbf{x})$ 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 $c_μ(q)$ has a nonnegative, log-concave coefficient sequence with interval support. Namely, this holds whenever $h$ is abelian and its complement-Ferrers partition $λ$ satisfies $\min\{λ_1,\ell(λ)\}\leq 3$; 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.
