2025/05/18 by Ayush Khaitan, Khaitan, Ayush, Ishan Mata +3
Mathematics · #05A20 (Primary) 05E05 #15A15 (Secondary) #Combinatorics (math.CO) #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Mathematical functions and polynomials #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2505.12178
openalex publication_date 2025/05/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 a1, …, an ∈ ℝ≥ 0, we prove that (1)/(n!) ∑π∈ Sn [∏_\i:i=π(i)\ ai] ≥ \frac1\binomn2 ∑_S ∈\binom[n]2 [ (∏_\i ∈ S\ ai )1/2], 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.