Elementary symmetric polynomials under the fixed point measure
Fuente:
arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866909614939832320 |
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| author | Khaitan, Ayush Mata, Ishan Narayanan, Bhargav |
| author_facet | Khaitan, Ayush Mata, Ishan Narayanan, Bhargav |
| contents | We identify a surprising inequality satisfied by elementary symmetric polynomials under the action of the fixed point measure of a random permutation. Concretely, for any collection of $n$ non-negative real numbers $a_1, \dots, a_n \in \mathbb{R}_{\geq 0}$, we prove that
\[
\frac{1}{n!} \sum_{π\in S_n} \left[\prod_{\{i:i=π(i)\}} a_i\right] \ge \frac{1}{\binom{n}{2}} \sum_{S \in\binom{[n]}{2}} \left[ \left(\prod_{\{i \in S\}} a_i \right)^{1/2}\right],
\]
and this bound is sharp. To prove this elementary inequality, we construct a collection of differential operators to set up a monotone flow that then allows us to establish the inequality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_12178 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Elementary symmetric polynomials under the fixed point measure Khaitan, Ayush Mata, Ishan Narayanan, Bhargav Combinatorics Differential Geometry 05A20 (Primary) 05E05, 15A15 (Secondary) We identify a surprising inequality satisfied by elementary symmetric polynomials under the action of the fixed point measure of a random permutation. Concretely, for any collection of $n$ non-negative real numbers $a_1, \dots, a_n \in \mathbb{R}_{\geq 0}$, we prove that \[ \frac{1}{n!} \sum_{π\in S_n} \left[\prod_{\{i:i=π(i)\}} a_i\right] \ge \frac{1}{\binom{n}{2}} \sum_{S \in\binom{[n]}{2}} \left[ \left(\prod_{\{i \in S\}} a_i \right)^{1/2}\right], \] and this bound is sharp. To prove this elementary inequality, we construct a collection of differential operators to set up a monotone flow that then allows us to establish the inequality. |
| title | Elementary symmetric polynomials under the fixed point measure |
| topic | Combinatorics Differential Geometry 05A20 (Primary) 05E05, 15A15 (Secondary) |
| url | https://arxiv.org/abs/2505.12178 |