{"id":"1317b1c2-6705-4357-aeb3-60f8401b7d90","arxiv_id":"2506.23082","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors introduce linked rook placements and prove a Hall-Littlewood expansion for chromatic quasisymmetric functions, with a corollary for unicellular LLT polynomials that is shown to be false by a small example.","lead":"This paper gives a new combinatorial formula for expressing the chromatic quasisymmetric function of a Dyck path in terms of Hall-Littlewood polynomials, using a new object called linked rook placements. It also claims a similar expansion for unicellular LLT polynomials, but that part appears to contain a mathematical error.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.8 is false for γ=N^2E^2 under the paper's own definitions; the proof introduces an invalid μ-dependent factor and misapplies the Carlsson–Mellit relation, leaving the advertised LLT expansion unsupported.","rationale":"The reader's weakest assumption is exactly the correctness of Proposition 2.5, and direct computation shows a concrete contradiction with the paper's own definitions for γ=N^2E^2. I reproduced this independently: from Definition 2.4, the LLT word sum is h_2+q e_2, while the chromatic quasisymmetric function from Definition 2.1 is (1+q)e_2; homogeneity of degree 2 makes the plethystic relation in Proposition 2.5 fail. The failure of Theorem 3.8 follows, and the proof's extra factor (1−q)^{n−ℓ(μ)} is not derived from Theorem 3.6; a homogeneous-degree count shows the scalar factors cancel globally. The main theorem is argued through the Abreu–Nigro modular-law criterion and does not directly use Proposition 2.5, so I do not claim the main Hall–Littlewood expansion itself is false; however, the abstract and title advertise the LLT application as a central contribution, and a demonstrably false secondary theorem with a flawed proof is sufficient to reject the paper as written. I found no machine-checked verification or reproducible code that would offset this internal inconsistency. A revised version might repair the normalization and the LLT statement, but the current text should not be accepted without substantial changes.","tokens_in":22889,"tokens_out":27993,"duration_ms":257627,"concrete_test":"Evaluate both sides of Theorem 3.8 for γ=N^2E^2, n=2, using only the definitions in Sections 2.5 and 2.6: compute X_γ=(1+q)e_2, LLT_γ via the word sum h_2+q e_2, and the proposed RHS via r_{γ,μ}, H_μ, and ω. The first identity gives h_2, the second gives e_2, while the left-hand side is h_2+q e_2, so the theorem fails. Equivalently, substitute these symmetric functions into Proposition 2.5 and check the plethysm LLT[(q−1)X]=(q−1)^2(h_2+q e_2); the identity fails unless h_2=e_2.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 3.8 is contradicted by a direct n=2 computation that uses only the paper's definitions. For γ=N^2E^2, Definition 2.1 gives X_γ(x;q)=(1+q)e_2(x), while Definition 2.4 gives LLT_γ(x;q)=Σ_{w∈Z_{>0}^2}q^{[w_1>w_2]}x^w=h_2(x)+q e_2(x). Since both are homogeneous of degree 2, plethysm with (q−1)X scales by (q−1)^2, so Proposition 2.5 would force X_γ=h_2+q e_2, not (1+q)e_2. Thus the quoted Carlsson–Mellit relation is not correct as stated for this path. Independently of that, the proof of Theorem 3.8 inserts the factor (1−q)^{n−ℓ(μ)} that does not follow from Theorem 3.6: writing [m]_q!=(q;q)_m/(1−q)^m and using homogeneity of P_μ gives a global cancellation of (q−1)^n, with no remaining μ-dependent power of (1−q). For γ=N^2E^2 the only μ is (1,1), so n−ℓ(μ)=0 and Theorem 3.8 predicts LLT=ω(H_{(1,1)})=h_2 in the first identity and e_2 in the second, whereas Definition 2.4 gives h_2+q e_2. The main theorem, Theorem 3.6, relies on the Abreu–Nigro criterion instead of Proposition 2.5 and may be salvageable, but the paper's second advertised result is demonstrably false; a correction would have to replace or repair both the proposition and Theorem 3.8.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces linked rook placements and defines a generating function r_{\\gamma,\\mu}(q) over them. It claims in Theorem 3.6 that the chromatic quasisymmetric function X_\\gamma(x;q) of a natural unit interval order has a Hall--Littlewood expansion whose coefficients are q^{area(\\gamma)-n(\\mu)} r_{\\gamma,\\mu}(q) \\prod_i [m_i(\\mu)]_q!. The proof follows the Abreu--Nigro modular-law criterion, with the bulk of the work in Lemmas 4.6 and 4.7, which are verified by diagrams. Applying the Carlsson--Mellit relation (Proposition 2.5), the paper then derives Theorem 3.8, a modified Hall--Littlewood expansion for unicellular LLT polynomials. The paper also gives a refinement of q-rook polynomials and poses several open problems.","tokens_in":23247,"tokens_out":60066,"duration_ms":570882,"significance":"If correct, the linked-rook description would be a natural q-analogue of the Stanley--Stembridge rook formula and would provide a new combinatorial model for Hall--Littlewood coefficients of chromatic quasisymmetric functions. The modular-law strategy is reasonable and the paper is self-contained. However, the manuscript contains internal counterexamples to its advertised theorems: Theorem 3.8 fails for a two-vertex path under the paper's own definitions, and a direct n=2 computation indicates that Theorem 3.6 itself is false as stated. The proofs are not machine-checked, and the main combinatorial lemmas rely on unformalized figure-based case checks. The claimed results therefore cannot be accepted in their current form.","major_comments":[{"comment":"Proposition 4.3 is false for n=2 under the definitions as written. For \\gamma=N^2E^2, the singleton linked rook ((1,2)) is a valid linked rook placement; its unique extension is the extended linked rook (1,1),(1,2),(2,2),(2,3), which covers both diagonal cells, so its type is (2). There are no cells above \\gamma, hence no free cells, and r_{\\gamma,(2)}=1. This directly contradicts the assertion in the proof of Proposition 4.3 that LRP(\\gamma,\\mu)=\\emptyset unless \\mu=(1^n). Substituting into (3.1), with area(N^2E^2)=1, n((2))=0, n((1,1))=1, P_{(2)}=h_2-q e_2 and [2]_q! P_{(1,1)}=(1+q)e_2, gives q(h_2-q e_2)+(1+q)e_2, whereas Definition 2.1 gives X_\\gamma=(1+q)e_2. Thus the main theorem fails for n=2 unless a different notion of type or extension is intended, which is not stated.","section":"Section 4, Proposition 4.3 and Theorem 3.6, Eq. (3.1)"},{"comment":"Theorem 3.8 is false for \\gamma=N^2E^2. Definition 2.4 gives LLT_\\gamma=h_2+q e_2 for this path, while Definition 2.1 gives X_\\gamma=(1+q)e_2. Since both are homogeneous of degree 2, the plethysm in (2.6) would force X_\\gamma=h_2+q e_2, a contradiction. Independently, the proof of Theorem 3.8 inserts the factor (1-q)^{n-\\ell(\\mu)} outside the sum over \\mu without justification: using [m]_q!=(q;q)_m/(1-q)^m and homogeneity of P_\\mu, the factors (q-1)^n and (q-1)^{-n} cancel globally, and no \\mu-dependent power of (1-q) remains. The statement is also ill-formed because the exponent n-\\ell(\\mu) depends on the summation variable \\mu. For n=2, the first identity predicts h_2 and the second predicts e_2, both disagreeing with Definition 2.4.","section":"Section 2.6 and Section 3, Proposition 2.5 and Theorem 3.8"},{"comment":"The proofs of Lemmas 4.6 and 4.7 are not formal. In each case, after defining a bijection, the equality of free-cell generating functions is asserted by reference to Figures 8--17 (e.g., \"by Figure 8, we have...\") without a written argument that cells away from the displayed local configuration are unaffected or that the displayed ranks determine the counts. Since these two lemmas constitute the entire proof of Proposition 4.1, the modular law for Y_\\gamma is not established as a verifiable proof. This is a rigor gap independent of the counterexamples above.","section":"Section 4.1, Lemmas 4.6 and 4.7"}],"minor_comments":[{"comment":"Equation (2.5) is written with the product \\prod_{i=1}^{\\ell(\\mu)}(q;q)_{m_i(\\mu)}; this indexing is inconsistent with the factors needed, since for \\mu=(3,1) it would omit the multiplicity m_3. The product should be over all i with m_i(\\mu)>0 or over all i\\ge 1. The identity should also be checked for consistency with (2.4), since for \\mu=(2) the two sides appear to differ.","section":"Section 2.3, Eq. (2.5)"},{"comment":"The first displayed equation in Theorem 3.8 places (1-q)^{n-\\ell(\\mu)} outside the summation over \\mu, but \\ell(\\mu) is not defined until the summation variable is specified; the factor must either be moved inside the sum or the theorem restated. The same issue appears in the proof.","section":"Section 3, Theorem 3.8"},{"comment":"There are several smaller presentation issues: the abstract and introduction refer to Theorem 1.4 while the body states the result as Theorem 3.8; the phrase \"row counted from the left\" in the definition of a_i(\\gamma) is ambiguous; and Definition 3.2 allows cells (i,j) with i\\le j but then places u_{2\\ell} in the extended row, whose coordinates have j=n+1, so the formal scope of j should be clarified.","section":"Throughout"}],"recommendation":"reject","confidential_remarks":"The stress-test note correctly identifies the failure of Theorem 3.8. In addition, I find a direct n=2 counterexample to the main theorem, Theorem 3.6, under the paper's own definitions: the singleton linked rook ((1,2)) on N^2E^2 has type (2), not type (1,1), and gives a nonzero contribution that the claimed formula cannot absorb. This suggests that the definitions of linked-rook type, extension, or free-cell count require substantive revision, and the central results are not reliable as stated. The paper should be rejected. If the authors believe my counterexample misreads an implicit convention, they should clarify the definitions; but as written, the claims are internally inconsistent."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper introduces a genuinely new object, linked rook placements, and aims to give a Hall-Littlewood expansion of chromatic quasisymmetric functions with coefficients r_{γ,μ}(q). The model is attractive and the q-rook refinement idea is worth pursuing, but the main theorem, Theorem 3.6, is false as stated.\n\nCounterexample: γ=N^2E^2, the path for the complete graph K_2. Under the paper's own definitions X_γ=(1+q)e_2. The board is the single cell (1,2). There is a linked rook placement of type (2) (one extended rook of length 2) with no free cells, so r_{γ,(2)}=1. With P_(2)=h_2−q e_2, the RHS of (3.1) gives q h_2+(1+q−q^2)e_2, not (1+q)e_2. Proposition 4.3's assertion that only μ=(1^n) contributes is plainly wrong for n=2.\n\nThe LLT results are also built on sand. Proposition 2.5 (Carlsson-Mellit) fails the same example: it would force X_γ=h_2+q e_2. The proof of Theorem 3.8 inserts a μ-dependent (1−q)^{n−ℓ(μ)} factor that does not come out of the algebra; for n=2 the theorem predicts h_2 and e_2, while direct computation gives h_2+q e_2.\n\nWhat does work: the authors have a real combinatorial idea, and the modular law framework of Abreu-Nigro is a legitimate strategy. Lemmas 4.6 and 4.7 are detailed, though the reliance on figures is a lesser concern. The citation of Griffin et al. is honest; the new part is the linked rook description, which is interesting even if currently incorrect.\n\nBottom line: this paper is not ready for publication. The errors are load-bearing and found in the smallest case. I would send it out for review because the model may be salvageable, but a rejection is the only defensible outcome. A revision should fix the definition of r_{γ,μ} or the exponents, and correct the CM relation.","headline":"The main Hall-Littlewood expansion is false for γ=N^2E^2; the linked rook model is promising but the paper's central theorems fail in the simplest case.","tokens_in":23817,"tokens_out":31349,"would_cite":false,"duration_ms":279432,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05A15","05A30"],"pacs":[],"model":"deepseek-v4-flash","headline":"Linked rook placements give a Hall–Littlewood expansion of chromatic quasisymmetric functions.","keywords":["chromatic quasisymmetric functions","Hall–Littlewood polynomials","linked rook placements","Dyck paths","unicellular LLT polynomials","modular law","free-cell statistic","q-rook polynomials"],"falsifier":"Compare both sides of Theorem 3.6 coefficient by coefficient in the Hall–Littlewood basis for a small non-complete Dyck path such as $\\gamma=N^3E^3$; any mismatch localizes an error in the modular-law bijections. For Theorem 3.8, evaluate the stated expansion at $\\gamma=N^2E^2$ and check whether the identity holds; this single path settles the validity of the plethystic application.","tokens_in":22651,"feed_emoji":"🏰","tokens_out":12500,"duration_ms":117323,"temperature":0.7,"pith_summary":"The paper aims to prove a new expansion: for a Dyck path γ, the chromatic quasisymmetric function $X_\\gamma(x;q)$ of the associated natural unit interval order is a sum over partitions μ of Hall–Littlewood polynomials $P_\\mu(x;q)$, with coefficients built from linked rook placements of type μ and a free-cell statistic. This is a $q$-analogue of the classical expansion of $X_\\gamma$ into monomial symmetric functions via ordinary rook placements, and it refines $q$-rook polynomials by recording the type of each placement. The proof shows that the proposed right-hand side obeys the modular law and a multiplicativity property, so a known uniqueness criterion for functions on Dyck paths forces it to equal $X_\\gamma$. A corollary applies the plethystic relation between $X_\\gamma$ and unicellular LLT polynomials to write unicellular LLT polynomials in modified transformed Hall–Littlewood bases; that second statement is more delicate and should be checked path by path.","feed_headline":"Rook placements count Hall–Littlewood coefficients","feed_subtitle":"A q-analogue of the classic rook formula: linked-rook placements weight the Hall–Littlewood expansion of X_γ.","key_machinery":"The central object is the linked rook placement: a set of chains of non-attacking rooks on the Ferrers board under a Dyck path, where consecutive rooks $(i,j)$ and $(j,k)$ in a chain share the middle coordinate. The extended linked rook placement adds one diagonal cell per chain and an extra row, and the rank of an extended rook in its column or row decides whether a cell above the path is free; a free cell is encoded as an fc-pair. The coefficient $r_{\\gamma,\\mu}(q)=\\sum_{P}q^{\\operatorname{fc}_\\gamma(P)}$ over placements of type μ is what carries the argument, since it supplies the Hall–Littlewood coefficients up to the factor $q^{\\operatorname{area}(\\gamma)-n(\\mu)}\\prod_i[m_i(\\mu)]_q!$. The proof mechanism is the modular-law criterion for functions on Dyck paths, together with explicit bijections that keep track of free-cell counts when rows or columns are exchanged, and a multiplicativity bijection for concatenating a Dyck path with $N^kE^k$.","core_discovery":"The central claim, stated as Theorem 3.6, is that $X_\\gamma(x;q)=\\sum_{\\mu\\vdash n} q^{\\operatorname{area}(\\gamma)-n(\\mu)}r_{\\gamma,\\mu}(q)\\left(\\prod_i [m_i(\\mu)]_q!\\right)P_\\mu(x;q)$, where $r_{\\gamma,\\mu}(q)$ is the generating function over linked rook placements of type μ weighted by the number of free cells. The linked rook placement is a chain of rooks $(a_1,b_1),\\ldots,(a_\\ell,b_\\ell)$ with $b_i=a_{i+1}$, and the free cells are defined through an extended placement that adds diagonal cells and an extra row; the statistics are tracked by ranks and fc-pairs. The paper proves the identity by verifying the modular law, multiplicativity under concatenation with $N^kE^k$, and the complete graph value $[n]_q!e_n$, and then invoking the modular-law criterion to conclude the right-hand side coincides with $X_\\gamma$. The $q=1$ specialization recovers the classical rook expansion, and the coefficients refine $q$-rook polynomials. A further theorem derives unicellular LLT expansions from the plethystic relation; the paper presents these as consequences of the same coefficients.","pith_inferences":["If the main expansion is correct for every Dyck path, the coefficients $r_{\\gamma,\\mu}(q)$ form a partition-type refinement of the $q$-rook polynomial, and the principal-specialization comparison suggests a natural refinement of $q$-hit polynomials that the paper does not construct.","The apparent failure of the LLT corollary on $\\gamma=N^2E^2$ indicates that the plethystic relation, or its application after substituting $X/(q-1)$, may require correction; because the main theorem is proved independently through the modular law, the two results stand or fall separately.","A direct bijection between linked rook placements and the pairs of P-tableaux and semistandard Young tableaux that appear in the Schur expansion would give a transparent proof of the Hall–Littlewood coefficients and would test whether the free-cell statistic has a simpler equivalent description."],"forward_implications":["At $q=1$ the formula recovers the ordinary rook-counting monomial expansion of $X_\\gamma$, so the main theorem is a genuine $q$-analogue of that classical identity.","Summing the linked-rook coefficients by length gives $R_{n-k}(\\gamma;q)=\\sum_{\\ell(\\mu)=k} r_{\\gamma,\\mu}(q)$, so the new coefficients refine $q$-rook polynomials by partition type.","The principal specialization $X_\\gamma(1,q,\\ldots,q^{\\alpha-1};q)=q^{\\operatorname{area}(\\gamma)}\\prod_i[\\alpha-a_i(\\gamma)]_q$ is recovered, connecting the coefficients to $q$-rook and $q$-hit polynomial data.","Where the plethystic relation is valid, the same linked-rook coefficients give a combinatorial description of unicellular LLT polynomials in the modified transformed Hall–Littlewood basis.","The modular-law proof provides an independent construction of a function on Dyck paths determined by complete-graph values, so the same coefficients can be tested against other expansions of chromatic quasisymmetric functions."],"supporting_citations":[{"why":"Supplies the modular-law criterion used to conclude that the proposed right-hand side equals $X_\\gamma$ after checking the required properties.","marker":"[AN21]"},{"why":"Gives the monomial expansion of $X_\\gamma$ by ordinary rook placements, the $q=1$ case that the main theorem refines.","marker":"[SS93]"},{"why":"Defines the chromatic quasisymmetric function and establishes symmetry for natural unit interval orders, fixing the object of study.","marker":"[SW16]"},{"why":"Provides the Hall–Littlewood polynomials, their Pieri rule, and the principal specialization identities used in the proofs.","marker":"[Mac95]"},{"why":"Supplies the plethystic relation between $X_\\gamma$ and unicellular LLT polynomials used to derive the LLT corollary.","marker":"[CM18]"},{"why":"Defines the $q$-rook polynomials that the new linked-rook coefficients refine.","marker":"[GR86]"}],"fun_headline_variants":["Linked rooks count Hall–Littlewood coefficients","Rook placements expand chromatic quasisymmetric polynomials","Hall–Littlewood from linked rook placements","Rook stats for LLT and Hall–Littlewood expansions","A rook formula for chromatic quasisymmetric functions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of the main expansion hangs on the modular-law and multiplicativity bijections for linked rook placements; separately, the LLT corollary hangs on the plethystic relation $X_\\gamma(x;q)=(q-1)^{-n}\\mathrm{LLT}_\\gamma[(q-1)X;q]$, whose application appears to break down for the Dyck path $\\gamma=N^2E^2$.","fun_headline_variants_meta":{"raw":{"variants":["Linked rooks count Hall–Littlewood coefficients","Rook placements expand chromatic quasisymmetric polynomials","Hall–Littlewood from linked rook placements","Rook stats for LLT and Hall–Littlewood expansions","A rook formula for chromatic quasisymmetric functions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000582,"raw_usage":{"total_tokens":2707,"prompt_tokens":878,"completion_tokens":1829,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":494,"completion_tokens_details":{"reasoning_tokens":1756}},"tokens_in":494,"tokens_out":1829,"duration_ms":14950,"temperature":1.0,"reasoning_tokens":1756,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T21:54:27.669788+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compare both sides of Theorem 3.6 coefficient by coefficient in the Hall–Littlewood basis for a small non-complete Dyck path such as $\\gamma=N^3E^3$; any mismatch localizes an error in the modular-law bijections. For Theorem 3.8, evaluate the stated expansion at $\\gamma=N^2E^2$ and check whether the identity holds; this single path settles the validity of the plethystic application.","supporting_citations":[],"review_version":1}