{"id":"729c4556-740e-47aa-bfc3-3442cb45a4ed","arxiv_id":"2502.05939","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Rook-Eulerian polynomials of Ferrers boards are real-rooted, with an interlacing refinement, and complete rook placements correspond to Bruhat lower intervals of 312-avoiding permutations.","lead":"This paper defines rook-Eulerian polynomials, which count one-rook-per-row placements on stair-shaped Ferrers boards, split by how many times a rook sits immediately to the right of the one above. It proves these polynomials always have only real roots, connecting complete rook placements to lower intervals of 312-avoiding permutations in the Bruhat order.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 12's matrix (5) computes the refined rook-Eulerian polynomials in the wrong order, so the interlacing induction as written proves a false statement.","rationale":"The central claim is real-rootedness of Qλ via interlacing of the refined sequence (4), and the proof's only mechanism is matrix (5) together with an appeal to [Brä15, Corollary 8.7]. The displayed matrix does not implement Lemma 10 in the order of the stated interlacing sequence; this is not merely a question about the external criterion's hypotheses but an internal orientation mismatch. A concrete small example shows the vector the matrix actually produces is not interlacing, so the proof as printed cannot be accepted. I still expect the theorem is true: the row-reversed staircase matrix is the natural switch operation and probably satisfies a suitable criterion, but the paper must state which orientation is intended and verify it against [Brä15, Corollary 8.7]. This is more specific than the reader's weakest_assumption, which concerned whether the criterion applies; the first obstacle is that the matrix in (5) is written in the wrong order. The reader's secondary flags, including the wrong product formula in Theorem 17 and the undocumented computer search, remain valid and reinforce a conditional/revise verdict rather than acceptance. I am not claiming the theorem is false; the example only shows that the displayed proof step is invalid as written.","tokens_in":10210,"tokens_out":33250,"duration_ms":296528,"concrete_test":"Recompute the recursion for λ+=(3,4,5,5,7,8) and λ=(3,4,4,6,7): express Qλ+_1,Qλ+_2,Qλ+_3 as linear combinations of (Qλ_3,Qλ_2,Qλ_1), form the coefficient matrix, and compare it with (5). The displayed matrix yields the vector (Qλ+_1,Qλ+_2,Qλ+_3); test interlacing of this vector by checking Qλ+_1⪯Qλ+_2, which fails because the relevant roots are approximately -0.05 and -0.174. Then form the row-reversed matrix and verify, using [Brä15, Corollary 8.7] or a direct interlacing check on further examples, whether that corrected matrix preserves the stated descending interlacing order.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Equations (3) and (5) are inconsistent with the order required for the interlacing induction. For λ=(3,4,4,6,7) and λ+=(3,4,5,5,7,8), Lemma 10 gives Qλ+_1=t(Qλ_1+Qλ_2+Qλ_3), Qλ+_2=Qλ_1+t(Qλ_2+Qλ_3), Qλ+_3=Qλ_1+Qλ_2+tQλ_3. With the right vector (Qλ_3,Qλ_2,Qλ_1), the coefficient matrix for the vector (Qλ+_3,Qλ+_2,Qλ+_1), which is the sequence Theorem 12 needs to prove interlacing, is [[t,1,1],[t,t,1],[t,t,t]]. The matrix displayed in (5), whose (i,j) entry is t for j≤λ1-(i-1), is the vertical reversal [[t,t,t],[t,t,1],[t,1,1]], and it computes (Qλ+_1,Qλ+_2,Qλ+_3). In this example (Qλ+_1,Qλ+_2,Qλ+_3) is not interlacing: after removing the common factor 3t^2, the third-largest roots of the polynomials are approximately -0.05, -0.174, and -0.5, so Qλ+_1⪯Qλ+_2 fails. Thus the displayed matrix proves a false statement, and the proof of the central theorem is not valid as written. The proof can likely be repaired by reversing the rows of Gλ, but then it must be checked that this corrected staircase matrix satisfies the hypotheses of [Brä15, Corollary 8.7]; the paper contains no such verification. The same indexing slip appears in the proof of Lemma 10, where the ascent drop is said to occur iff j≥i−1, although the correct condition, matching formula (3), is j≥i.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":10659,"tokens_out":8426,"duration_ms":70804,"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":[{"comment":"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":"Section 2, Eq. (5) and proof of Theorem 12"},{"comment":"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.","section":"Section 3.1, Theorem 17"},{"comment":"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.","section":"Sections 3.1, 3.2, 3.3"}],"minor_comments":[{"comment":"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":"Section 2, Lemma 10 proof"},{"comment":"The phrase 'by an identical argument as in the proof of Proposition 12' refers to Theorem 12, not Proposition 12.","section":"Section 2, Theorem 13"},{"comment":"The definition of Q_π(t) writes t^{des(π)} instead of t^{des(σ)}; the same typo appears in Proposition 22.","section":"Section 3.2, Definition before Example 18"},{"comment":"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":"Section 2, Eq. (5)"},{"comment":"The notation I(s)_n in the paragraph after Definition 14 should be I^s_n for consistency with the definition.","section":"Section 3.1, Definition 14"}],"recommendation":"major_revision","confidential_remarks":"The paper is clearly written and the combinatorial framework is appealing, but the indexing error in the proof of Theorem 12 is load-bearing and must be fixed, together with a verification that the corrected matrix satisfies Brändén's criterion. The authors should also make the computer searches reproducible and correct the evaluation error in Theorem 17. If these issues are addressed, the paper could be suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my read.\n\nThe paper introduces rook-Eulerian polynomials Qλ for Ferrers boards, proves real-rootedness via an interlacing induction, adds a same-phase stability extension, and separates this family from s-Eulerian polynomials. The Bruhat interpretation (Proposition 3) is folklore but cleanly proved. If the main theorem is up after repairs, this is a useful subfield contribution.\n\nThe soft spots are real. The matrix in equation (5) does not do what the proof needs. Lemma 10's recurrence in (3) is correct, but with the right-hand vector (Qλ_λ1,...,Qλ_1), the displayed Gλ computes (Qλ+_1,...,Qλ+_m) in ascending order. The theorem needs the descending sequence. For λ=(3,4,4,6,7) and λ+=(3,4,5,5,7,8), the ascending sequence (Qλ+_1,Qλ+_2,Qλ+_3) is not interlacing, so as written the induction proves a false statement. Reversing the rows of Gλ gives the right order, but then the appeal to [Brä15, Cor. 8.7] is an unverified one-liner; the corrected staircase matrix should satisfy it, but the paper should show it. The same indexing bug appears in Lemma 10's proof: 'j≥i−1' should be 'j≥i'.\n\nTheorem 17 is also troubled. The evaluation at t=1 states ∏s_i = ∏(λ_i - n + i), but for λ=(2,3,5,5,5) the right-hand product is 0 while Qλ(1)=24. The correct count is ∏(λ_i - i +1)=24. The exhaustive computer search is undocumented, so as it stands the theorem is unproved. This is a secondary claim, but it should be fixed.\n\nMinor: Theorem 13 says 'as in Proposition 12' but there is no Proposition 12; and the base case in Theorem 12 is n=1 despite the n>1 statement.\n\nNet: the central idea is sound and the repair looks local, but the written proof is not currently valid. This deserves a serious referee, not a desk reject, because the mistakes are identifiable and likely typos rather than conceptual failures. I would not cite it in its current form.","headline":"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.","tokens_in":11169,"tokens_out":7616,"would_cite":false,"duration_ms":67572,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05A15","26C10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that every rook-Eulerian polynomial of a Ferrers board is real-rooted, via an interlacing argument.","keywords":["Ferrers boards","rook placements","Eulerian polynomials","real-rootedness","interlacing sequences","Bruhat order","312-avoiding permutations","same-phase stability"],"falsifier":"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.","tokens_in":10046,"feed_emoji":"♜","tokens_out":8011,"duration_ms":64444,"temperature":0.7,"pith_summary":"The paper introduces rook-Eulerian polynomials, which count row-complete rook placements on Ferrers boards by ascents, and proves that every such polynomial is real-rooted. The argument refines each polynomial according to the first rook's column, shows the refined sequence is interlacing, and applies a recurrence that preserves interlacing via a known criterion for staircase matrices. A second result identifies the complete rook placements on a Ferrers board with the lower Bruhat interval of a 312-avoiding permutation, giving the polynomials a Catalan-flavored interpretation. The paper also proves a multivariate same-phase stability result, shows the family is distinct from the s-Eulerian polynomials, and explores variants for descents and excedances, including counterexamples to real-rootedness in general Bruhat intervals.","feed_headline":"Rook-Eulerian polynomials are real-rooted on all Ferrers boards","feed_subtitle":"A proof by interlacing shows these ascent-counting polynomials have only real roots, with a Bruhat-interval interpretation.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the interlacing-preservation criterion (Corollary 8.7) on which the induction step of Theorem 12 relies.","marker":"[Brä15]"},{"why":"Classical result that the full symmetric-group Eulerian polynomial is real-rooted; underlies the base case of the family.","marker":"[Fro10]"},{"why":"Shows Q_λ(t) is the q=1 specialization of a permanent-based polynomial t^{-1}A_F(t,q), placing rook-Eulerian polynomials in prior rook theory.","marker":"[SY12]"},{"why":"Proves that Bruhat intervals correspond to rook placements on skew Ferrers boards, used for the 312-avoiding interpretation.","marker":"[Sjö07]"},{"why":"Introduces s-Eulerian polynomials and proves their real-rootedness via a similar interlacing recurrence; the paper compares to show the families are distinct.","marker":"[SV15]"},{"why":"Established real-rootedness of multiset Eulerian polynomials for rectangular Ferrers boards; the paper conjectures a Ferrers-board generalization.","marker":"[Sim84]"}],"fun_headline_variants":["Rook-Eulerian: real roots for every Ferrers board","Interlacing proves rook-Eulerian polynomials real-rooted","All Ferrers boards: rook-Eulerian polynomials have real roots","New proof: rook-Eulerian polynomials are real-rooted","Generalizing Eulerian: rook polynomials real-rooted"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Rook-Eulerian: real roots for every Ferrers board","Interlacing proves rook-Eulerian polynomials real-rooted","All Ferrers boards: rook-Eulerian polynomials have real roots","New proof: rook-Eulerian polynomials are real-rooted","Generalizing Eulerian: rook polynomials real-rooted"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000918,"raw_usage":{"total_tokens":3859,"prompt_tokens":787,"completion_tokens":3072,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":403,"completion_tokens_details":{"reasoning_tokens":2986}},"tokens_in":403,"tokens_out":3072,"duration_ms":20294,"temperature":1.0,"reasoning_tokens":2986,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T17:19:07.684433+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}