Elementary symmetric polynomials under the fixed point measure
Ayush Khaitan, Ishan Mata, Bhargav Narayanan
TL;DR
This work studies an inequality for elementary symmetric polynomials under the fixed-point measure of a random permutation. For a nonnegative vector $\mathbf{a}=(a_1,\dots,a_n)$, the paper proves $ \mathbb{E}_{\pi\in S_n}\left[\prod_{i\in \mathrm{fix}(\pi)} a_i\right] \ge \mathbb{E}_{S\in \binom{[n]}{2}}\left[\left(\prod_{i\in S} a_i\right)^{1/2}\right]$, and reformulates it as $\sum_{k=0}^n d(n,k) s_k(\mathbf{a}) \ge s_2(\sqrt{\mathbf{a}})$ using rencontres numbers $D(n,k)$. The left-hand side equals $\mathrm{per}(M(\mathbf{a}))/n!$, connecting the inequality to matrix permanents. The authors develop a novel monotone-flow proof by defining $L_n(\mathbf{x})$ and $R_n(\mathbf{x})$, analyzing $f_n(\mathbf{x})=L_n(\mathbf{x})-R_n(\mathbf{x})$, and employing differential operators $\mathcal{O}_k$ to show a unique interior minimum at $\mathbf{1}$ with $f_n(\mathbf{1})=0$, thereby establishing the inequality for $n\ge4$ (with elementary handling of $n\le3$). This approach provides a new tool for proving inhomogeneous inequalities for symmetric polynomials and suggests directions for extending algebraic inequality theory beyond homogeneous settings, including potential applications to permanents and related combinatorial structures.
Abstract
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.