bibkey: alexanderssonjalquemener2025rook authors: Per Alexandersson, Aryaman Jal, and Maena Quemener year: 2025 title: “Real-rootedness of rook-Eulerian polynomials” doi: 10.48550/arXiv.2502.05939 url: https://arxiv.org/abs/2502.05939v1 claim: “Conjecture 28: Ferrers-board multiset rook-Eulerian real-rootedness and pairwise interlacing of the decreasing first-letter refinements.” strata_touched:

• D5/S3/Combinatorics/Permutation/MultisetRookEulerianInterlacingRefutation license: citation-only triage: anchor

Multiset rook-Eulerian polynomials

The primary source is arXiv:2502.05939v1, Section 3.3, pages 11–12. Conjecture 28, page 12, reads:

For Ferrers boards, the polynomial $R(\lambda ,\alpha ;t)$ is real-rooted. Moreover, $R_{{\lambda }_1}(\lambda ,\alpha ;t),R_{{\lambda }_1-1}(\lambda ,\alpha ;t),\dots ,R_2(\lambda ,\alpha ;t),R_1(\lambda ,\alpha ;t)$ forms an interlacing sequence.

Definition 8, page 6, reads:

Let $f$ and $g$ be polynomials with positive leading coefficients and real, non-positive zeros, $a_i$ and $b_i$, respectively. We say that $f$ interlaces $g$, and we write $f⪯g$ if $\cdots \le a_3\le b_3\le a_2\le b_2\le a_1\le b_1\le 0$. Note that $\mathrm{deg}(f)=\mathrm{deg}(g)$ or $\mathrm{deg}(f)+1=\mathrm{deg}(g)$.

Definition 9, page 6, reads:

A sequence $F=(f_1,\dots ,f_n)$ of real-rooted polynomials is interlacing if $f_i⪯f_j$ for $1\le i<j\le n$.

Section 3.3, page 11, defines the words:

Let $\alpha =({\alpha }_1,\dots ,{\alpha }_k)$ be non-negative integers with total sum $n$, and let $\lambda$ and $\mu$ be integer partitions such that ${\lambda }_i>{\mu }_i$ for all $i$. We let $\mathcal{W}(\lambda /\mu ,\alpha )$ be all words with ${\alpha }_i$ entries equal to $i$, such that ${\mu }_i<w_i\le {\lambda }_i$ for all $i=1,2,\dots ,n$.

For Ferrers boards, $\mu =0$. Equations (10) and (11) define $R={\sum }_{w\in \mathcal{W}}{t}^{\mathrm{asc}(w)}$ and $R_j={\sum }_{w\in \mathcal{W},w_1=j}{t}^{\mathrm{asc}(w)}$. Section 2, page 4, specifies:

if $w_1{w}_2\dots w_{\ell }$ is a word, then $i$ is an ascent of the word if $w_i<w_{i+1}$.

Examples 25–26 give twelve fitting rearrangements of $11223$ on board $22233$, with polynomial $t^3+8t^2+3t$. The source cites Simion for rectangular-board real-rootedness and Ma–Pan, Theorem 1.11, for interlacing in the rectangular case. Conjecture 28 concerns arbitrary Ferrers boards.

The six-row board $(3,3,3,3,4,4)$ with content $(2,2,1,1)$ refutes the interlacing conjunct: the required bottom comparison is $-7\le -4-\sqrt{13}$. The universal real-rootedness conjunct remains open.

Verified locator

• DOI: https://doi.org/10.48550/arXiv.2502.05939
• URL: https://arxiv.org/abs/2502.05939v1
• Version and location: arXiv:2502.05939v1; Section 2, page 4 for ascents; Definitions 8–9, page 6 for interlacing; Section 3.3, page 11 for words, equations (10)–(11) and Examples 25–26; Conjecture 28, page 12.