A common interleaver for two antichain polynomials on $[k]$$\times$ $P_{n,s}$

Abstract

The first author and Dong \cite{DD} proposed three conjectures on antichain generating polynomials. Jiang \cite{Jiang} recently proved Conjectures 4.3 and 4.5, concerning real-rootedness and \(γ\)-positivity. We prove Conjecture 4.2 by adapting his method from \([k]\times [2] \times [n]\) to \([k]\times P_{n,s} \), where \(P_{n,s}\) is a two-row Ferrers shape. We establish real stability for a family of bivariate polynomials associated with adjacent shapes, then apply the Chudnovsky-Seymour compatibility criterion to obtain a common interleaver. As noted in \cite{DD}, Conjecture 4.2 also implies Conjecture 4.3.