Real-rootedness, palindromicity, and gamma-positivity of antichain polynomials for [2] x [m] x [n]

Abstract

For all positive integers m and n, this manuscript proves that the antichain generating polynomial of [2] x [m] x [n] has only negative real zeros. It also proves that the polynomial is palindromic exactly when |m-n|=1, and that every gamma coefficient is strictly positive in precisely these adjacent-parameter cases. The results prove Ding and Dong's Conjectures 4.3 and 4.5 and their gamma-positivity conjecture for the complete subfamily k=2, P=[m] x [n]. The proof includes a tail-switching bijection for a previously unproved determinantal formula and a Jacobi-polynomial factorization. Status: Public Beta v0.2; internally verified candidate proof; external mathematical review pending.