{"id":"hexagon/2610.00720","pdf":"https://hexagonmath.org/pdf/2610.00720v1","url":"https://hexagonmath.org/abs/2610.00720v1","tags":["math.CO","msc:05E05","msc:33C45","msc:26D07","Meixner polynomials","Schur polynomial","Turan ratio","homogeneous symmetric polynomial","monotonicity"],"score":0,"title":"Monotonicity of Turán-ratios of symmetric polynomials","domain":"math.CA","origin":"ai-generated","source":"hexagon","authors":["Suvrit Sra"],"reviews":0,"subject":"math.CA","summary":"We study the Turán-ratio $S_{\\lambda,i}(x)^2/(S_{\\lambda,i-1}(x)S_{\\lambda,i+1}(x))$, where $S_{\\lambda,i}(x):=s_\\lambda(1^{[n-i]},x^{[i]})$, and $s_\\lambda$ is the Schur polynomial in $n$ variables (indexed by partition $\\lambda$) with $i$ arguments equal to $x$ and $n-i$ of them equal to $1$. Specifically, we prove that this Turán-ratio is strictly increasing for $x>1$ and $1\\le i<n$ (unless all $n$ padded parts of $\\lambda$ are equal, then it is identically one). For one-row shapes, our result answers a question of Ait-Haddou about complete homogeneous symmetric polynomials. For this…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["GPT Pro"],"discussed":0,"published":"2026-11-02","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"all"}