Real-rootedness and interlacing for multiset rook-Eulerian polynomials

Establish that for every Ferrers board \(\lambda\) and every non-negative integer content vector \(\alpha\), the multiset rook-Eulerian polynomial \(R(\lambda,\alpha;t)\) is real-rooted, and prove that the sequence \(R_{\lambda_1}(\lambda,\alpha;t),R_{\lambda_1-1}(\lambda,\alpha;t),\ldots,R_2(\lambda,\alpha;t),R_1(\lambda,\alpha;t)\) of first-entry refinements forms an interlacing sequence.

Background

The paper generalizes rook-Eulerian polynomials to words of prescribed content fitting inside Ferrers boards, allowing multiple rooks in a single column. The resulting polynomial $R(\lambda/\mu,\alpha;t)$ records the ascent statistic on these constrained multiset words.

The conjecture extends Simion’s real-rootedness theorem for rectangular Ferrers boards and incorporates an interlacing refinement related to later work on multiset Eulerian polynomials. The authors state it for Ferrers boards, while explicitly observing that real-rootedness fails for certain skew Ferrers boards.