{"id":"b28375b9-edaf-448c-9039-3b80e95c17c7","arxiv_id":"1908.08161","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For 2-hook partitions, divided symmetrizations over graphs containing the path expand into Schur functions with coefficients given by a new combinatorial construction and an arc-breaking rule.","lead":"This paper gives a step-by-step recipe for writing a certain symmetric polynomial, defined from a partition and a graph, as a sum of standard simple pieces. The recipe covers a special family of partitions and works for any graph in a broad class, going beyond previous results.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"2-hook construction omits the trivial term t = λ_n + ω_n, so Theorem 3.2/Proposition 3.1 cannot be correct as stated.","rationale":"The reader's weakest assumption concerned unverified α-correspondences in the induction for Theorem 3.2. My review finds a more concrete and more fundamental problem: the 2-hook construction is not merely under-verified; it is incomplete. The element t = λ_n + ω_n belongs to N_{λ_n,E_n} by Lemma 2.1 with the empty justifying pair set, and it contributes the Schur function s_{λ_n} to the expansion. Since the construction always outputs a permutation of [n] with maximum entry n, it cannot generate this element for n ≥ 3. The proof of Theorem 3.2 only treats the permutation case, so the completeness half of the theorem is false as stated. This directly undermines the paper's advertised constructive method: the first construction does not find all terms of the expansion. The final formula in Theorem 4.3 might be salvageable by adding the trivial s_{λ_n} term separately and revising Theorem 3.2 to describe only the nontrivial permutations, but the current version contains a false central claim. I therefore recommend moving from CONDITIONAL to REJECT, while acknowledging that a careful revision could restore correctness.","tokens_in":15732,"tokens_out":28829,"duration_ms":259654,"concrete_test":"Compute the divided symmetrization for n = 3, λ = (2,1,0), and G = P_3 with edges (1,2),(2,3) directly from the definition. The numerator polynomial is x_1^3 x_2^2 (x_1 − x_3), whose divided symmetrization equals s_(2,1,0) − s_(1,1,1). The 2-hook construction from the initial state (3,0,0) yields only the final permutation (3,2,1), i.e., the s_(1,1,1) term, and cannot produce t = (4,2,0) = λ_3 + ω_3, which is the unique justifying pair set for the s_(2,1,0) term. This missing term settles the concern.","verdict_should_be":"REJECT","load_bearing_attack":"Proposition 3.1 and Theorem 3.2 claim that the 2-hook construction generates exactly the set N_{λ_n,E_n}. But t = λ_n + ω_n is in N_{λ_n,E_n}: in Lemma 2.1 it is justified by the empty pair set (coefficient +1), and it has all distinct entries. For n ≥ 3, its first coordinate is n+1. The 2-hook construction, however, starts from (n,0,…,0) and places the values n,n−1,…,1, so every completed construction is a permutation of [n]; it can never produce an entry larger than n. Thus t = λ_n + ω_n is never generated. The proof of Theorem 3.2 explicitly restricts to vectors t that form a permutation of [n] with t_1 = n and t_n = 1, so the non-permutation case is never addressed. Consequently the claimed completeness of the construction is false, and the constructive method as stated omits the s_{λ_n} term (coefficient +1) from the expansion in Theorem 4.3.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the divided symmetrization operator for partitions λ of length n and graphs G containing a path, focusing on the 2-hook shape λ_n=(2,1,...,1,0). It defines a set N of exponent vectors that can occur in the expansion and claims a combinatorial construction, the '2-hook construction', that generates exactly these vectors. A strengthened version, the 'strong 2-hook construction', is supposed to produce, for each such vector, a justifying pair set used to compute the coefficient. The main result, Theorem 4.3, gives a Schur-function expansion of the divided symmetrization in which the index set is N and each coefficient is an alternating sum over justifying pair sets obtained by arc breaks over marked indices. Section 5 gives a recurrence and a closed formula for the size of N.","tokens_in":15920,"tokens_out":17794,"duration_ms":169833,"significance":"If the main claims are correct, the paper supplies a fully combinatorial, constructive description of the Schur expansion for a previously unsolved family of divided symmetrizations. The intended mechanism—a state construction with associated pair sets plus an arc-breaking rule for recovering all justifications—is natural and would be a genuine extension of Postnikov's and Petrov's results. The statements are concrete and falsifiable, and the running example illustrates the intended combinatorics clearly. However, the manuscript as written has not secured these claims: the definition of the set being enumerated is internally inconsistent, the construction demonstrably misses a term under the literal reading of Lemma 2.1, and several load-bearing inductive verifications are asserted rather than shown.","major_comments":[{"comment":"The construction does not enumerate the set defined in the paper. Under Lemma 2.1, the tuple λ_n+ω_n is justified by the empty pair set and has distinct coordinates, so it belongs to N_{λ_n,E_n}; it also has empty justifying pair set whenever the empty set is allowed in the coefficient sum. But the 2-hook construction starts from (n,0,...,0), never changes the first coordinate, and places the values n,n-1,...,1, so every completed tuple is a permutation of [n]. Since λ_n+ω_n has first coordinate n+1, it can never be produced. The proof of Theorem 3.2 explicitly begins with vectors t for which t_1,...,t_n form a permutation of [n] with t_1=n and t_n=1, thereby excluding this case. Thus Proposition 3.1 and Theorem 3.2 are false as stated. If the intended object is only the nontrivial 2-hook permutations, that should be stated, and the omitted s_{λ_n} term must be added separately in Theorem 4.3; if the intended object is the full N, the construction is incomplete.","section":"Sections 2 and 3"},{"comment":"The set N_{λ,E} is not well defined because the exponent formulas are inconsistent. The rewriting immediately before equation (1) gives terms of the form λ+o_{E(G)}+v(E) with E ⊆ E_n-E(G), while Lemma 2.1 writes λ+ω_n+v(E). These are not the same: for n=3 and G=P_3, λ+o_G=(3,2,0) but λ+ω_n=(4,2,0). Since every subsequent statement about which tuples are '2-hook permutations' depends on the exponent convention, the paper must state which definition of N is being used and reconcile Lemma 2.1 with the preceding derivation.","section":"Section 2"},{"comment":"The induction proving completeness of the strong 2-hook construction depends on four asserted α-correspondences between states on n and states on n-1. These are introduced with the phrase 'Basic calculations using our rules show...' and none of the four correspondences is verified; the α-correspondence must simultaneously match the sets of available indices, the marking status, the vectors s and M, and the subsequent legal placements. Because the correctness of the induction and of the completeness argument rests on these assertions, the verification cannot be omitted. Please provide explicit checks or a standalone lemma with proof.","section":"Section 3"},{"comment":"The proof that every justifying pair set is obtainable by arc breaks over marked indices is incomplete in the case t_k=n-1 with k>2. The construction of E^b_A and E^b_B, the claim that these justify t_A and t_B, and the transfer of the arc-break sequences back to E_a are all asserted rather than proved; the notation is also inconsistent, since E^a_A is defined using E instead of E_a. Since Theorem 4.3's coefficient formula relies on this statement, the proof needs to be completed or the statement reorganized as a separate lemma with a full argument.","section":"Section 4"}],"minor_comments":[{"comment":"The condition defining α-correspondence for states repeats the same expression on both sides: it reads |[n_a-1]-I(p_{S_a})| = |[n_a-1]-I(p_{S_a})|; the second term should be |[n_b-1]-I(p_{S_b})|.","section":"Section 3"},{"comment":"In the rules for unmarked placements, the text says 'in the state S′ formed by a legal placement on index k and marking the index' in both the marked and unmarked cases; the second occurrence should say 'unmarking the index'.","section":"Section 3"},{"comment":"The bullet condition 'If x_i^S = x_o^S and k = y_i^S and k = y_i^S' contains a duplicated clause; it should presumably be 'k = y_i^S' only.","section":"Section 3"},{"comment":"The displayed justifying pair set in the running example lists (2,5) twice; a set should be written without repetition.","section":"Section 3"},{"comment":"The index set in Theorem 4.3 is written as N_{λ_n,E_n}, which is independent of G, while the coefficient sum depends on G through E(G); the surrounding text works with N_{λ_n,E_n-E(G)}. Please clarify the intended index set. The same ambiguity affects Lemma 5.1, where N_{λ_n,E_n} is counted as if it were large, although the literal definition of N_{λ,E_n} with E=E_n gives only the singleton {λ_n+ω_n}.","section":"Theorem 4.3"}],"recommendation":"major_revision","confidential_remarks":"The paper's notation for N is inconsistent in ways that go beyond typographical slips: the definition of the set, the statement of Lemma 2.1, the contents of Corollary 2.4, and the enumeration in Section 5 cannot all be correct simultaneously. An editor may wish to ask the author to rewrite the setup with a single unambiguous convention before further review. The underlying construction idea is plausible and worth pursuing, but the current version is not yet a reliable proof of the main theorem."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"I read the Ince paper on divided symmetrizations for 2-hooks. The main idea—two constructive algorithms, one for the index set and one for the coefficients—is attractive, and the paper does a good job of laying out the reduction from the divided symmetrization to a Schur expansion via justifying pair sets. The arc-breaking procedure in Section 4 is a genuinely useful way to think about coefficient multiplicities, and the enumeration recurrence in Section 5 is a nice addition.\n\nBut there is a problem that, as far as I can tell, invalidates the central claim. The paper's own Lemma 2.1 says that any t with a justifying pair set and distinct entries lies in N_{λ_n,E_n}. Take t = λ_n + ω_n. It has distinct entries, and the empty pair set justifies it, so it is in N. The 2-hook construction, however, starts at (n,0,...,0) and places the values n,n-1,...,1, so it can only produce permutations of [n]. The vector λ_n+ω_n has first entry n+1 and last entry 0, so the construction never reaches it. Consequently Proposition 3.1 and Theorem 3.2, which claim the construction generates exactly N, are false. The proof of Theorem 3.2 explicitly restricts to permutations with t_1=n and t_n=1, so the exceptional case is never handled. Theorem 4.3 then misses the s_{λ_n} term, which has coefficient 1.\n\nThis is a genuine gap, not a cosmetic one. I suspect the authors meant to exclude the exceptional vector (Corollary 2.4 appears to have a typo, saying 't=λ_n' where it must mean 't=λ_n+ω_n'), but the paper as written is internally inconsistent: Lemma 2.1 includes it, Corollary 2.4 tries to leave it out, and the construction never addresses it.\n\nOn the softer side, the inductive proofs rely on 'basic calculations' that are not shown, particularly the alpha-correspondences in Theorem 3.2. That is a verifiability issue, but it's secondary; even if those calculations check out, the exceptional term still breaks the main theorem.\n\nMy recommendation: send it to peer review, but tell the authors to fix the exceptional case. The arc-breaking ideas are worth preserving, and once the λ_n+ω_n term is added (or explicitly excluded with a correct argument), the paper could be a solid contribution. As it stands, the main result is not correct.","headline":"The main completeness theorem is false: N contains λ_n+ω_n by the paper's own definition, but the 2-hook construction never produces it, so Theorem 4.3 misses the s_{λ_n} term.","tokens_in":16440,"tokens_out":10188,"would_cite":false,"duration_ms":83061,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E05","05E10","05A15"],"pacs":[],"model":"deepseek-v4-flash","headline":"Divided symmetrizations of 2-hook partitions over path graphs have a Schur expansion whose terms and coefficients are produced by two finite combinatorial constructions.","keywords":["divided symmetrization","Schur functions","2-hook partitions","combinatorial construction","arc breaking","Kostka numbers","enumerative combinatorics"],"falsifier":"For a fixed small $n$, such as $n=5$ or $6$, enumerate all tuples $t$ for which some pair set $E\\subseteq E_n$ satisfies $\\lambda_n+\\omega_n+v(E)=t$ with distinct coordinates, and compare that list with the outputs of the strong 2-hook construction; the paper's claim fails if the lists differ. A second check would test Theorem 4.2 by searching for a justifying pair set for a 2-hook permutation that cannot be obtained from the placement solution by arc breaks restricted to marked indices.","tokens_in":15501,"feed_emoji":"🧩","tokens_out":6204,"duration_ms":55356,"temperature":0.7,"pith_summary":"This paper studies the divided symmetrization $\\overrightarrow{D}_{\\lambda,G}$, a signed sum over permutations of a monomial divided by products $(x_{\\delta(i)}-x_{\\delta(j)})$ over edges of a graph $G$, and tries to give a complete Schur-function expansion in the special case where $\\lambda$ is a 2-hook partition $(2,1,\\dots,1,0)$. The paper claims that every term in the expansion is indexed by a 2-hook permutation, an object generated by a simple placement construction, and that every coefficient is a signed count of certain pair sets obtained by breaking arcs over marked indices. If correct, this turns an algebraic expansion into finite combinatorial data and extends the known evaluations of divided symmetrizations for paths and trees to a full coefficient formula. A sympathetic reader would care because the construction is explicit enough to compute coefficients by hand for small $n$ and to count the number of terms exactly.","feed_headline":"Combinatorial formula found for 2-hook divided symmetrizations","feed_subtitle":"Coefficients come from signed counts of arc-breakable pair sets built by a placement construction.","key_machinery":"The load-bearing objects are 2-hook permutations: integer $n$-tuples $t$ whose coordinates are a permutation of $\\{1,\\dots,n\\}$ with $t_1=n$ and $t_n=1$, obtainable by satisfying $\\lambda_n+\\omega_n+v(E)=t$ for some pair set $E$. The strong 2-hook construction builds each such tuple by placing decreasing values on legal indices while carrying a pair set, and the induction relating states on $n$ and states on $n-1$ runs through $\\alpha$-correspondence, a bijection between available indices that preserves the quantities $s$ and $M$ controlling legality. The coefficient computation then uses arc breaking: replacing pairs $(a,c)$ by $(a,b),(b,c)$ over a marked index $b$, which preserves the vector sum $v(E)$ and therefore justifies the same $t$. This identity, $v(E_a)=v(E_b)$, is what lets the paper sum justifying pair sets by sign without recomputing the algebraic expansion.","core_discovery":"The central result, Theorem 4.3, states that for the 2-hook $\\lambda_n=(2,1,\\dots,1,0)$ and any path graph $G$ on $n$ vertices, $\\overrightarrow{D}_{\\lambda,G}$ equals $$\\sum_{t\\in N_{\\lambda_n,E_n}}\\left(\\sum_{E\\in \\mathcal{E}_t \\cap \\mathcal{P}(E_n-E(G))}(-1)^{|E|}\\right)s_{l(t)},$$ where $N_{\\lambda_n,E_n}$ is the set of 2-hook permutations, $\\mathcal{E}_t$ is the set of all justifying pair sets for $t$, and $l(t)$ is the partition obtained by sorting the coordinates of $t$ and shifting by $(n-1,n-2,\\dots,0)$. The paper proves that the strong 2-hook construction generates exactly the 2-hook permutations and that every justifying pair set can be reached from the construction's placement solution by breaking arcs only over marked indices. This yields the first complete combinatorial description of the Schur expansion for this family of divided symmetrizations.","pith_inferences":["The paper explicitly notes that a 3-hook has multiple pair sets that cannot be reduced to a single placement solution, so the arc-breaking mechanism appears special to 2-hooks; a natural extension would be to identify the largest family of partitions for which a marked-arc reachability description still exists.","A testable extension is to turn the $\\alpha$-correspondence induction into a recursive algorithm that generates coefficients in time proportional to the number of 2-hook permutations, which the recurrence suggests grows roughly like $(2+\\sqrt{2})^n$.","The signed sum over arc-breakable pair sets resembles an evaluation of a graph invariant, so one could ask whether the coefficient of $s_{(2,1^{n-2})}$ in $\\overrightarrow{D}_{\\lambda_n,G}$ has a direct interpretation as a Tutte-polynomial or matching-polynomial evaluation."],"forward_implications":["For every path graph $G$, the Schur expansion of $\\overrightarrow{D}_{\\lambda_n,G}$ is determined by the finite list of 2-hook permutations, so computing the expansion reduces to enumerating those permutations and their marked-arc breakings.","Each coefficient is a signed count of justifying pair sets reachable by arc breaks over marked indices; this replaces algebraic expansion with a purely combinatorial sign sum.","The number of terms satisfies $N_n=a(n-3)$ with $a(0)=1$, $a(1)=2$, $a(n)=4a(n-1)-2a(n-2)$, giving an exact count of the 2-hook permutations for each $n$.","Because every $t\\in N_{\\lambda_n,E_n}$ other than $\\lambda_n+\\omega_n$ has $l(t)$ a hook-shaped partition, the Schur functions appearing in the expansion are only of hook shape."],"supporting_citations":[{"why":"Supplies the definition of divided symmetrization, the evaluation $D_{\\lambda,P}$ for paths, and the polytope background that motivates the problem.","marker":"[1]"},{"why":"Gives the combinatorial interpretation for evaluations over arbitrary trees, the direct precedent the paper extends.","marker":"[2]"},{"why":"Provides explicit computations with divided symmetrization used as context for the open cases.","marker":"[3]"}],"fun_headline_variants":["Signed arc-breakable pair sets give 2-hook expansion","Constructive method yields Schur coefficients for 2-hook symmetrizations","First complete Schur expansion for 2-hook divided symmetrizations","Placement construction computes 2-hook symmetrization coefficients","Signed pair sets resolve 2-hook divided symmetrization expansion"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the four $\\alpha$-correspondences asserted in the proof of Theorem 3.2, said to follow from basic calculations that are not shown, are all correct for every reachable state; if any one fails, the strong 2-hook construction could miss legitimate 2-hook permutations or include invalid ones, and the same inductive link would damage Theorem 4.2.","fun_headline_variants_meta":{"raw":{"variants":["Signed arc-breakable pair sets give 2-hook expansion","Constructive method yields Schur coefficients for 2-hook symmetrizations","First complete Schur expansion for 2-hook divided symmetrizations","Placement construction computes 2-hook symmetrization coefficients","Signed pair sets resolve 2-hook divided symmetrization expansion"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000447,"raw_usage":{"total_tokens":2210,"prompt_tokens":852,"completion_tokens":1358,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":468,"completion_tokens_details":{"reasoning_tokens":1265}},"tokens_in":468,"tokens_out":1358,"duration_ms":11233,"temperature":1.0,"reasoning_tokens":1265,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:49:10.428204+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a fixed small $n$, such as $n=5$ or $6$, enumerate all tuples $t$ for which some pair set $E\\subseteq E_n$ satisfies $\\lambda_n+\\omega_n+v(E)=t$ with distinct coordinates, and compare that list with the outputs of the strong 2-hook construction; the paper's claim fails if the lists differ. A second check would test Theorem 4.2 by searching for a justifying pair set for a 2-hook permutation that cannot be obtained from the placement solution by arc breaks restricted to marked indices.","supporting_citations":[{"cited_title":"Permutohedra, associahedra, and beyond","cited_arxiv_id":null,"evidence_quote":"Supplies the definition of divided symmetrization, the evaluation $D_{\\lambda,P}$ for paths, and the polytope background that motivates the problem."},{"cited_title":"Combinatorial and Probabilistic Formula e for Divided Symmetrization","cited_arxiv_id":null,"evidence_quote":"Gives the combinatorial interpretation for evaluations over arbitrary trees, the direct precedent the paper extends."},{"cited_title":"Explicit Computations with the Di vided Symmetrization Operator (2014) 15","cited_arxiv_id":null,"evidence_quote":"Provides explicit computations with divided symmetrization used as context for the open cases."}],"review_version":1}