1-Rayleigh property of the normalized Kahn–Saks polynomial

Prove that the normalized Kahn–Saks polynomial associated with every chain \(x_1\leq\cdots\leq x_k\) in a poset is 1-Rayleigh; namely, establish the stated Rayleigh inequalities for all admissible multi-indices \(\alpha,i,j\) and every nonnegative evaluation point \((u,v)\).

Background

The normalized Kahn–Saks polynomial encodes the numbers of linear extensions of a poset subject to prescribed gaps along a fixed chain. The paper formulates the 1-Rayleigh property as a stronger conjecture than the generalized cross-product conjecture and proves that the 1-Rayleigh conjecture would imply the latter. Thus, the conjecture remains unresolved within the paper.

References

Let us formulate an even stronger version of this conjecture. \begin{conjecture} \label{con:kahn-saks-rayleigh} The normalized Kahn-Saks polynomial for a chain $x_1 \leq \ldots \leq x_k$ is $1$-Rayleigh, i.e.

\partial\alpha N() \partial{\alpha +e_i +e_j} N()(u,v) \leq \partial{\alpha+e_i} N() \partial{\alpha +e_j} N()(u,v)

or every $(u,v) \in R_{\geq 0}k$. \end{conjecture}

— Diagonalizations of denormalized volume polynomials  (2502.13305 - Ross et al., 18 Feb 2025) in Conjecture~\ref{con:kahn-saks-rayleigh}, Section 3, Application to poset inequalities