Research Preprints · Combinatorics
Monotonicity of Turán-ratios of symmetric polynomials
Abstract
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 case, we also offer a second proof through the zeros of Meixner polynomials that gives an explicit positive formula for the logarithmic derivative. We conclude with a Turán-ratio conjecture for Macdonald polynomials.
Versions
Version numbers are this site's own; arXiv numbering, where present, is separate. Every released file carries an OpenTimestamps proof made on its release day.
| Version | Released | Length | File | Proof | arXiv |
|---|---|---|---|---|---|
| v1 | 7 October 2026 | 10 pp | .ots | — |
Cite
@misc{sra2026sym-turan-v1,
author = {Suvrit Sra},
title = {{Monotonicity of Turán-ratios of symmetric polynomials}},
year = {2026},
month = oct,
howpublished = {Preprint, \url{https://github.com/suvrit/preprints/tree/main/preprints/sym-turan/v1}},
note = {Version v1, released 7 October 2026},
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pip install opentimestamps-client curl -O https://suvrit.github.io/papers/sym-turan/sym-turan-v1.pdf -O https://suvrit.github.io/papers/sym-turan/sym-turan-v1.pdf.ots ots verify sym-turan-v1.pdf.ots