REVIEW 3 major objections 5 minor 1 cited by
Real-rootedness of rook-Eulerian polynomials
T0 review · 3 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash
Pith's one-line read The paper proves that every rook-Eulerian polynomial of a Ferrers board is real-rooted, via an interlacing argument.
desk verdict Real-rootedness is likely true, but the written proof has a row-order error that makes the main induction invalid as stated, and Theorem 17's product formula is wrong; worth a referee after fixes. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The key machinery is the refined polynomial family Q_λ^i(t), counting row-complete rook placements whose first rook sits in column i, together with the recurrence Q_{λ+}^i = ∑_{j<i} Q_λ^j + t ∑_{j≥i} Q_λ^j (Lemma 10). The induction organizes the refined vector into an m×λ1 matrix product with the staircase matrix G_λ, defined by g_{i,j}=t for j≤λ1−(i−1) and 1 otherwise. Because G_λ satisfies the hypotheses of the interlacing-preservation criterion [Brä15, Corollary 8.7], multiplying an interlacing vector by G_λ yields an interlacing vector, which moves the property from the smaller board to the larger one.
What would settle it
For a small Ferrers board, say λ=(2,3,3), compute the refined polynomials $Q_λ^{3}$, $Q_λ^{2}$, $Q_λ^{1}$ and check whether their roots alternate in the interlacing order; a failure would disprove Theorem 12. Alternatively, test directly whether multiplying an interlacing vector of polynomials by G_λ for some λ yields a non-interlacing vector, which would show the claimed preservation step fails.
Extended reading notes
Core claim
The central claim is Theorem 12: for any Ferrers board λ=(λ1,…,λn) with n>1, the polynomials $Q_λ^{{λ1}}$, $Q_λ^{{λ1−1}}$, …, $Q_λ^{1}$ form an interlacing sequence, so the rook-Eulerian polynomial Q_λ(t)=∑_σ $t^{{asc(σ)}}$ has only real, non-positive roots. The proof is by induction on n: for a board extended by adding a row and shifting columns, the refined polynomials are obtained by multiplying the refined vector of a smaller board by a staircase matrix G_λ whose entries are t on the left part of each row and 1 on the right; this matrix is shown to preserve the interlacing property using the criterion of [Brä15, Corollary 8.7]. Consequently the multivariate rook-Eulerian polynomial is same-phase stable, and the univariate polynomials are real-rooted for all Ferrers boards.
Load-bearing premise
The induction step relies on the assertion, not verified in the paper, that the staircase matrix G_λ meets the conditions of the interlacing-preservation criterion [Brä15, Corollary 8.7] for the specific row and column order used in equation (4); if that criterion does not apply to matrices of this exact shape and orientation, the interlacing induction collapses.
Editorial extensions
If this is right
- Every rook-Eulerian polynomial of a Ferrers board has real, non-positive roots, so its coefficients form a log-concave sequence.
- For every 312-avoiding permutation π, the ascent-generating polynomial of the Bruhat interval [id,π]_B is real-rooted, yielding a new family of real-rooted polynomials indexed by 312-avoiding permutations.
- The multivariate rook-Eulerian polynomial is same-phase stable, meaning that every restriction to a positive ray is real-rooted.
- The rook-Eulerian polynomials are not contained in the family of s-Eulerian polynomials, so they constitute a genuinely new generalization of the classical Eulerian polynomials.
- Descent-generating polynomials of arbitrary Bruhat intervals and of weak-order intervals are not real-rooted in general; the paper gives explicit 7-element and 17-element counterexamples, so the 312-avoiding condition is essential.
Reading between the lines
- The interlacing structure suggests that the rook-Eulerian polynomials might be moment sequences of probability distributions, analogous to the way classical Eulerian polynomials give rise to log-concave distributions; this could be explored via the roots.
- The correspondence with 312-avoiding permutations indicates that other Catalan objects, such as Dyck paths or non-crossing partitions, may carry analogous real-rooted ascent polynomials, possibly admitting a similar staircase-matrix argument.
- If the conjectured ultra-log-concavity of weak-order interval descent polynomials holds, it would refine Brenti's conjecture by giving a stronger coefficient property for this class of intervals.
- A direct testable extension is to check whether the staircase matrix G_λ preserves stability for multivariate refinements beyond same-phase stability, which could yield stable multivariate rook-Eulerian polynomials.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper defines rook-Eulerian polynomials Q_λ(t) for Ferrers boards λ as ascent-generating polynomials over row-complete rook placements, and proves (Theorem 12) that the refined polynomials Q_λ^i form an interlacing sequence, hence Q_λ is real-rooted. It also gives a multivariate same-phase stability result (Theorem 13), a Bruhat-order interpretation for complete rook placements on 312-avoiding permutations (Proposition 3), a comparison with s-Eulerian polynomials (Theorem 17), and several conjectures and counterexamples about descents and excedances in Bruhat and weak order intervals.
Significance. If the main theorem is correct, the paper provides a clean generalization of Eulerian polynomials with an elegant proof via Brändén's matrix interlacing criterion, and the connection to lower Bruhat intervals of 312-avoiding permutations is attractive. The paper is careful to give a deletion bijection for the main recurrence, and the multivariate same-phase stability statement is a natural strengthening. However, the proof of the central real-rootedness theorem contains a serious indexing error in the matrix formulation, and several secondary claims rely on undocumented computer calculations. The main result is not established as written, though the error appears repairable.
major comments (3)
- [Section 2, Eq. (5) and proof of Theorem 12] The matrix-vector equation is indexed incorrectly. With the left vector written as (Q^{λ+}_m, ..., Q^{λ+}_1)^T and the entry rule g_{i,j}=t for j ≤ λ1-(i-1), the top row (all t's) computes Q^{λ+}_1, not Q^{λ+}_m. For λ=(3,4,4,6,7), λ+=(3,4,5,5,7,8), Lemma 10 gives the coefficient matrix for the vector (Q^{λ+}_3, Q^{λ+}_2, Q^{λ+}_1) with right vector (Q^λ_3, Q^λ_2, Q^λ_1) as [[t,1,1],[t,t,1],[t,t,t]], whereas the matrix displayed in (5) is its vertical reversal [[t,t,t],[t,t,1],[t,1,1]]. The latter proves interlacing of the ascending tuple (Q^{λ+}_1, Q^{λ+}_2, Q^{λ+}_3), which is false in this example (after removing a common factor, the third-largest roots are approximately -0.05, -0.174, -0.5). Thus the induction as written proves a false statement. The proof can likely be repaired by reversing the rows of G_λ, but then one must verify that the corrected staircase matrix satisfies the hypotheses of [Brä15, Corollary 8.7]; the paper contains no such verification.
- [Section 3.1, Theorem 17] The evaluation at t=1 is incorrect. The paper states that Q_λ(1) equals ∏_{i=1}^n (λ_i - n + i), but for λ=(2,3,5,5,5) the right-hand side is zero while Q_λ(1)=24. The correct number of row-complete placements is ∏_{i=1}^n (λ_i - i + 1). Since the reduction to a finite search rests on this false identity, and since the claimed exhaustive computer search is not documented (no algorithm or code), Theorem 17 is not established as written.
- [Sections 3.1, 3.2, 3.3] Several claims depend on undocumented computer calculations: the exhaustive search in Theorem 17, the counterexample in Example 18, the polynomial in Proposition 22, and the computation in Remark 29. For a proof-based combinatorics paper, these should be accompanied by reproducible code or a precise description of the finite verification so that the reader can check them.
minor comments (5)
- [Section 2, Lemma 10 proof] The sentence 'the number of ascents in the rook placement on λ decreases by 1 if and only if j≥i−1' should read 'j≥i' to match the recurrence (3).
- [Section 2, Theorem 13] The phrase 'by an identical argument as in the proof of Proposition 12' refers to Theorem 12, not Proposition 12.
- [Section 3.2, Definition before Example 18] The definition of Q_π(t) writes t^{des(π)} instead of t^{des(σ)}; the same typo appears in Proposition 22.
- [Section 2, Eq. (5)] The display in (5) is internally inconsistent: the bottom row is drawn with all t's, whereas the stated entry rule gives only the first λ1-m+1 entries of that row as t.
- [Section 3.1, Definition 14] The notation I(s)_n in the paragraph after Definition 14 should be I^s_n for consistency with the definition.
Circularity Check
No circularity: the main real-rootedness proof derives a recurrence from a deletion bijection and applies an external interlacing criterion; self-citations are background only.
full rationale
Theorem 12 is proved by induction using Lemma 10's recurrence, which is derived directly from a deletion bijection on rook placements: 'the operation of deleting the first row of λ+ followed by the ith column of λ bijectively maps rook placements on λ+ with the first rook in column i, to rook placements on λ.' The induction step expresses this recurrence through the matrix Gλ and invokes [Brä15, Corollary 8.7], an external published criterion, to conclude that interlacing is preserved. This is not a self-citation chain and does not define the target polynomials in terms of the desired conclusion. The refined polynomials Qλ_i are defined by conditioning on the first column entry, and their sum Qλ is real-rooted as a direct consequence of interlacing; no parameter is fitted to data and no prediction is renamed from an input. The only self-citations, [AN21] and [AJ24], appear in background or conjectural discussion in Section 3 and are not load-bearing for the central theorem. No definitional, fitted-input, or imported-uniqueness circularity is present. There may be an independent question about whether the displayed matrix satisfies the hypotheses of [Brä15] in the stated orientation, but that is a correctness and verification issue, not a circularity issue.
Assumptions & free parameters
assumptions (1)
- standard math The staircase matrix Gλ satisfies the hypotheses of [Brä15, Corollary 8.7] in the order used in (4).
Cite this review
Pith. "Pith review of Real-rootedness of rook-Eulerian polynomials." pith.science (2026). https://pith.science/paper/TQRHAIZ6
@misc{pith2026250205939,
author = {Pith},
title = {Pith review of: Real-rootedness of rook-Eulerian polynomials},
year = {2026},
howpublished = {\url{https://pith.science/paper/TQRHAIZ6}},
note = {Machine review of arXiv:2502.05939}
}
abstract
We introduce rook-Eulerian polynomials, a generalization of the classical Eulerian polynomials arising from complete rook placements on Ferrers boards, and prove that they are real-rooted. We show that a natural context in which to interpret these rook placements is as lower intervals of $312$-avoiding permutations in the Bruhat order. We end with some variations and generalizations along this theme.
Figures
Forward citations
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Reference graph
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Reviewed August 8, 2026 · model on record in the stance chip above.
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