Proof of the monotone column permanent conjecture

TL;DR

This paper proves the stability of a certain matrix permanent polynomial, confirming a conjecture and deriving implications for real-rootedness, Eulerian polynomials, and inequalities.

Contribution

It establishes the stability of the permanent of a matrix involving a diagonal and all-ones matrix, resolving a recent conjecture and deriving multiple mathematical consequences.

Findings

Proved the stability of per(J_nZ_n + A) in variables z_i.

Showed per(zJ_n + A) has only real roots as a polynomial in z.

Derived new permanental inequalities and applications to Eulerian polynomials.

Abstract

Let A be an n-by-n matrix of real numbers which are weakly decreasing down each column, Z_n = diag(z_1,..., z_n) a diagonal matrix of indeterminates, and J_n the n-by-n matrix of all ones. We prove that per(J_nZ_n+A) is stable in the z_i, resolving a recent conjecture of Haglund and Visontai. This immediately implies that per(zJ_n+A) is a polynomial in z with only real roots, an open conjecture of Haglund, Ono, and Wagner from 1999. Other applications include a multivariate stable Eulerian polynomial, a new proof of Grace's apolarity theorem and new permanental inequalities.