{"id":"6a951c42-9270-40ba-af5e-5d3e1ddeb7dd","arxiv_id":"1908.04164","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A permutation is 1432-avoiding if and only if its double Grothendieck polynomial equals a signed sum over set-valued Rothe tableaux of its Rothe diagram.","lead":"This paper introduces set-valued Rothe tableaux, a new combinatorial object for permutations, and proves they generate double Grothendieck polynomials exactly for the family of 1432-avoiding permutations. This gives a compact tableau formula for a broad new class of K-theoretic Schubert polynomials.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.10 is only sketched and the displayed equations (2.9)/(2.29) mix D(w) with D(wsr); the induction in Theorem 2.11 depends on the omitted row-r+1 case analysis.","rationale":"The reader's verdict is CONDITIONAL and identifies Theorem 2.10's sketched proof as the weakest assumption. My reading agrees: the induction in Theorem 2.5/2.11 is the load-bearing step, and it depends on exact class-polynomial identities whose row-r+1 case is omitted. I also noticed that equations (2.9) and (2.29) mix D(w) and D(wsr), which makes the omitted bookkeeping harder to verify; this reinforces the reader's concern rather than changing it. I did not find a concrete counterexample or an internal inconsistency in the parts that are proved in detail: Lemma 2.12 is correct, the low-degree argument for the 1432-containing direction is sound because distinct labelings give distinct monomials, and the base case for w0 is correct. Therefore the appropriate verdict remains CONDITIONAL: accept only after the missing case analysis is completed or replaced by a fully detailed proof.","tokens_in":20506,"tokens_out":27189,"duration_ms":268125,"concrete_test":"Write out a complete proof of Theorem 2.10 for the row r+1 case, deriving (2.24) without invoking Theorem 2.7, and checking equation (2.29) against the true shapes D(wsr) and D(w). Independently, compute both sides of Theorem 1.1 for all permutations in S_6 using the divided-difference definition of G_w and exhaustive enumeration of SVRT(w,f0); any mismatch would disprove the statement, while a clean pass would confirm that the unproved case analysis is the only obstacle.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central theorem is proved by induction through Theorem 2.11, which asserts π_r G_{w s_r}(C;x,y) = G_w(Φ(C);x,y). That equality requires the class formulas Theorem 2.7 and Theorem 2.10 to match, including the exact y-exponents and the contribution h(C',r+1;x,y) of the squares in P(T',r+1). Theorem 2.10 is not proved: it is dismissed with 'the proof is nearly the same as the arguments for Theorem 2.7,' and the only stated difference is exactly the row r+1 case. This is not a formality, because D(w) and D(wsr) differ precisely near rows r and r+1: the square (r,w_r) is deleted and row-r squares are shifted down to row r+1. One must verify that the non-r/non-r+1 entries in shifted squares re-index under m_{ij} exactly as the h(C',r+1;x,y) factor claims, and that no configuration in this row escapes the P/Q analysis used for i>r+1. The concern is compounded by notational incoherence: (2.9) defines the class sum for SVRT(wsr,f0) with a product over D(w), and (2.29) evaluates a tableau T of shape D(wsr) on D(w). As written, (2.29) is ill-defined. If the intended bookkeeping at the deleted square is not supplied, or if the index identity ℓ_r(T)=ℓ_{r+1}(T')−1 is off by one in this special row, then π_r G_{w s_r}(C;x,y) need not equal G_w(Φ(C);x,y), and the induction proving Theorem 2.1 collapses. The external Lemma 2.12 is quoted correctly, but it amplifies rather than removes the need for the missing case analysis.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces set-valued Rothe tableaux, a filling model for the Rothe diagram of a permutation, and proposes that for a 1432-avoiding permutation w the double Grothendieck polynomial G_w(x,y) equals the sign-weighted sum over flagged set-valued Rothe tableaux given in (1.2). The converse is also claimed: if w contains a 1432 pattern, the formula fails. The proof of the forward direction proceeds by induction on length using the isobaric divided difference operator π_r, where r is the first ascent of w; the proof is organized around an equivalence relation on tableaux and a bijection between equivalence classes of SVRT(w s_r, f0) and SVRT(w, f0). The converse is proved by passing to single Schubert polynomials and using the balanced-labeling model of Fomin, Greene, Reiner, and Shimozono. A final section uses Knutson–Miller–Yong tableau complexes to derive two alternative formulas for G_w(x,y).","tokens_in":20899,"tokens_out":12984,"duration_ms":115769,"significance":"If the main theorem is correct, the paper provides the first tableau formula for double Grothendieck polynomials of 1432-avoiding permutations, a class equinumerous with vexillary permutations, and it specializes to Matsumura's formula for 321-avoiding permutations. The strategy of using equivalence classes to make the tableau model compatible with divided differences is natural and, in outline, sound; the paper also gives a useful connection to tableau complexes and to the balanced-labeling model for the converse statement. However, the current version of the manuscript does not contain a complete proof of a central lemma (Theorem 2.10), and several displayed formulas have indexing/domain problems that are load-bearing for the induction step. The result is plausible and significant, but the proof needs substantial repair before the claims are fully supported.","major_comments":[{"comment":"Theorem 2.10, the formula for G_w(C';x,y) for an equivalence class C' in SVRT(w,f0), is only sketched: the text says 'The proof is nearly the same as the arguments for Theorem 2.7' and that the only difference is that P(T',r) is empty and P(T',r+1) contributes h(C',r+1;x,y). This is load-bearing because Theorem 2.11, specifically equation (2.32), uses this exact factor to identify π_r of the r-row contribution with -h(C',r+1;x,y), and Theorem 2.11 is the induction step establishing Theorem 2.5 and hence Theorem 2.1. The omitted row r+1 case is precisely where D(w) and D(w s_r) differ: the square (r,w_r) is deleted and the r-row squares to its right are shifted down to row r+1. That is exactly the configuration that cannot be assumed to follow from the analysis for i>r+1. A complete proof of Theorem 2.10, including verification of the y-exponents in h(C',r+1;x,y) and of the first factor in (2.24), is required for the induction to go through.","section":"Section 2.2, Theorem 2.10"},{"comment":"Equations (2.9) and (2.29) are not well-defined as written. In (2.9), G_{w s_r}(C;x,y) is defined with a product over (i,j)∈D(w), but the tableaux T∈C have shape D(w s_r), so T(i,j) is undefined for squares of D(w) that are not in D(w s_r). Similarly, the left-hand side of (2.29) multiplies over (i,j)∈D(w) with T∈SVRT(w s_r,f0), and the right-hand side multiplies over D(w) with T'∈SVRT(w,f0); neither product is meaningful without an explicit identification of the square sets. This is not a cosmetic issue: (2.29) is substituted into (2.28), and the comparison between the two sides is the step that transfers the non-r/non-r+1 contribution from w s_r to w after the geometric shift. The intended index sets (presumably D(w s_r) on the left, together with the explicit matching of shifted squares) must be stated, and the equality proved for each type of square, especially the shifted row-r squares.","section":"Section 2.2, equations (2.9) and (2.29)"},{"comment":"The definition of ℓ_i(T) in (2.10) uses m_{i p_i}(w s_r) for every tableau T, but T may belong to SVRT(w,f0) as well as SVRT(w s_r,f0). For T∈SVRT(w,f0), the leftmost square (r+1,p_{r+1}) of P(T,r+1) is typically not a square of D(w s_r), so m_{i p_i}(w s_r) is undefined. This matters because the polynomial h(C',r+1;x,y) in Theorem 2.10 and the equality ℓ_r(T)=ℓ_{r+1}(T')-1 in (2.32) depend on this quantity. The authors should either define ℓ_i(T) separately for the two Rothe diagrams, with m computed in the corresponding diagram, or prove that the two definitions agree on the relevant squares.","section":"Section 2.2, equation (2.10)"}],"minor_comments":[{"comment":"There are several typographical slips, including 'Bu ch' in the abstract, 'shew diagram' after Figure 2.1, and 'there exits' in the proof of Theorem 2.13; these should be corrected.","section":"Abstract and Introduction"},{"comment":"The proof of Lemma 2.9 refers to 'the following two claims' and proves them, but the final sentence of the proof relies on the flag condition T(t,k)={t} for 1≤t≤i; this implication could be made more explicit for the reader.","section":"Section 2.2, Lemma 2.9"},{"comment":"The proof of (2.17) invokes Figure 2.5 and says 'we see that (2.17) holds'; adding a short explanation of how the definition of m_{i,j}(w s_r) in (1.1) yields the consecutive exponents ℓ_r(T)+1, ..., ℓ_r(T)+b_r(T) would improve readability.","section":"Section 2.2, proof of (2.17)"},{"comment":"The substitution in the proof of Theorem 1.4, replacing t_{x/arrownot↦a} by x_a y_{m_{ij}(w)+i-a} and then replacing x_t by 1-x_t and y_t by 1/(1-y_t), is stated very briefly; a few words on how the substitution acts on the K-polynomial and on the expression (x_t ⊕ y_{m}) would help the reader verify the formulas (1.3) and (1.4).","section":"Section 3, Theorem 3.1"}],"recommendation":"major_revision","confidential_remarks":"The central idea and the overall strategy are credible, and the paper addresses a natural class of permutations that is equinumerous with vexillary permutations. However, the gap in Theorem 2.10 is not a small omission: it is the very place where the geometries of D(w) and D(w s_r) differ, and the induction proof depends on it. The undefined products in (2.9) and (2.29) reinforce the concern that the row-shift bookkeeping has not been worked out in the written proof. I would like to see a revised version with a full proof of Theorem 2.10 and with all index sets made explicit; if that can be done, the paper is likely to be a valuable contribution to the tableau-formula literature."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe main claim of Fan--Guo is that the double Grothendieck polynomial of w has a set-valued Rothe tableau formula if and only if w avoids 1432. That is new and, as far as I can tell, correct in its overall architecture. It extends Matsumura's formula for 321-avoiding permutations, and the converse via balanced labelings is a crisp argument. No free parameters, no circularity; the induction over the first ascent is the natural strategy and matches what Wachs and Matsumura did for adjacent pattern classes.\n\nWhat is actually new: set-valued Rothe tableaux as a model, the exact pattern-class characterization, and the two alternative formulas via tableau complexes. The proof of Theorem 2.1 is substantial and mostly careful. Lemmas 2.8 and 2.9 look right, and the balanced-labeling construction in Theorem 2.13 checks out. The specialization to Matsumura is well explained and is a good sanity check.\n\nSoft spots, in proportion. The biggest is Theorem 2.10: it is only a sketch, with the one stated difference being the squares in P(T', r+1) and the factor h(C', r+1; x, y). That is exactly what Theorem 2.11 needs when it trades the row-r factor against h(C', r+1) via (2.32). If the row-r+1 case behaved differently from the rows i > r+1, the induction would not go through. I do not see an actual error, but the sketch is load-bearing, and a referee should ask for the missing case analysis to be written out.\n\nThere are also two display-level problems. In (2.9), G_{ws_r}(C;x,y) is defined with a product over D(w) and m_{ij}(w), which is inconsistent with the tableaux having shape D(ws_r). It should use D(ws_r) and m_{ij}(ws_r). Similarly, (2.29) has products over D(w) with T(i,j) on the left, where T has shape D(ws_r); as written, that left side is not well-defined. The intended bookkeeping after the bijection Φ moves row-r squares down to row r+1 and probably makes the equality true, but the display should be rewritten.\n\nThese issues do not convince me the main theorem is false. The notation problems look like typos from comparing the two shapes, not a gap in the mathematical strategy. Together with the sketch of Theorem 2.10, though, they mean the paper is not yet in clean publishable form. A serious referee can sort this out.\n\nWho is this for? People working in combinatorial Schubert calculus and K-theoretic tableau formulas. If the sketch gets filled in, this is a solid addition to the literature. I would send it to a competent referee rather than desk reject; the result is worth having and the missing piece is finite case analysis, not a new idea.","headline":"A genuinely new tableau model and an iff result for 1432-avoiding permutations, with one load-bearing lemma left as a sketch and a couple of notational slips that need cleaning up.","tokens_in":21431,"tokens_out":5144,"would_cite":true,"duration_ms":53529,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E10","05E05","14M15"],"pacs":[],"model":"deepseek-v4-flash","headline":"The double Grothendieck polynomial has a Rothe-tableau formula exactly for 1432-avoiding permutations.","keywords":["set-valued Rothe tableaux","double Grothendieck polynomials","1432-avoiding permutations","Rothe diagrams","isobaric divided difference operators","tableau complexes","Schubert polynomials","pattern avoidance"],"falsifier":"Expanding both sides of (1.2) for the 1432-avoiding, non-321-avoiding permutation $w=35142$ and comparing coefficients would test the forward claim: equality on every monomial is exactly what Theorem 2.1 predicts, while any mismatch disproves it. For the reverse claim, the same coefficient comparison for $w=1432$ should exhibit at least one monomial where the two sides differ, since Theorem 2.2 asserts that this obstruction always exists.","tokens_in":20297,"feed_emoji":"🧮","tokens_out":15691,"duration_ms":143915,"temperature":0.7,"pith_summary":"This paper introduces set-valued Rothe tableaux, which fill the Rothe diagram of a permutation with finite nonempty sets of positive integers so that rows are weakly decreasing and columns are strictly increasing. It argues that the double Grothendieck polynomial of a permutation, a polynomial representative for the K-theory class of a Schubert variety, is generated by these tableaux exactly when the permutation avoids the pattern 1432. For 1432-avoiding permutations the polynomial is the signed sum specified in Theorem 1.1; for every other permutation the paper proves that the same sum fails. The result matters because explicit tableau formulas for Grothendieck polynomials are known only for special permutation families, and this paper adds a new family while specializing to the known 321-avoiding tableau formula. Two further tableau formulas are derived from the tableau-complex viewpoint.","feed_headline":"1432-avoiding permutations get a Rothe-tableau formula","feed_subtitle":"This tableau sum matches the double Grothendieck polynomial for 1432-avoiding permutations and fails outside them","key_machinery":"The load-bearing object is the set-valued Rothe tableau: a filling of the Rothe diagram $D(w)$ with finite nonempty subsets of positive integers in which rows are weakly decreasing and columns strictly increasing under the set order $A<B$ when $\\max A<\\min B$ and $A\\leq B$ when $\\max A\\leq\\min B$. The proof mechanism is an induction on length using the isobaric divided difference operator $\\pi_r$, where the two tableau sets $\\mathrm{SVRT}(ws_r,f_0)$ and $\\mathrm{SVRT}(w,f_0)$ are partitioned into equivalence classes supported on the squares containing $r$ or $r+1$; a bijection between class sets and a factorization of the class contributions show $\\pi_r G_{ws_r}(C;x,y)=G_w(\\Phi(C);x,y)$. The reverse direction uses the balanced-labeling model of Schubert polynomials to produce a labeling of $D(w)$ that is not a single-valued Rothe tableau. A secondary mechanism, the tableau complex of the cited reference [17], converts the main formula into two alternative tableau formulas.","core_discovery":"The central discovery is an exact characterization. A permutation $w$ is 1432-avoiding if and only if $$G_w(x,y)=\\sum_{T\\in \\mathrm{SVRT}(w,f_0)}(-1)^{|T|-\\ell(w)}\\prod_{(i,j)\\in D(w)}\\prod_{t\\in T(i,j)}\\bigl(x_t\\oplus y_{m_{ij}(w)+i-t}\\bigr),$$ where $D(w)$ is the Rothe diagram of $w$, $\\mathrm{SVRT}(w,f_0)$ is the set of set-valued Rothe tableaux of shape $D(w)$ flagged by $f_0=(1,2,\\ldots,n)$, $\\ell(w)$ is the inversion length, $m_{ij}(w)$ is the number of diagram squares in row $i$ at or to the left of column $j$, and $a\\oplus b=a+b-ab$. The forward direction is proved by induction along the first ascent using the isobaric divided difference operator, and the reverse direction is proved by showing that a 1432 pattern forces a failure already in the single-variable Schubert polynomial obtained from the lowest-degree part.","pith_inferences":["An outside reader may test whether the same equivalence-class induction, which uses only the first-ascent condition and local configurations of squares, extends to other single-pattern avoidance classes beyond 1432; the paper does not make this claim.","Because the paper notes the equal enumeration of 1432-avoiding and 2143-avoiding permutations, one may look for a direct statistic-preserving bijection between set-valued Rothe tableaux of 1432-avoiding permutations and flagged set-valued Young tableaux of 2143-avoiding permutations; the paper does not construct one.","The tableau-complex formulas suggest a purely topological check: proving that the Rothe tableau complex is shellable would give an independent, noncomputational confirmation of the K-polynomial identities, a question not addressed here."],"forward_implications":["Setting all $y_i$ to zero turns the main formula into a signed tableau sum for the single Grothendieck polynomial of every 1432-avoiding permutation.","Taking the lowest-degree homogeneous part and replacing $y_i$ by $-y_i$ yields the corresponding formulas for double and single Schubert polynomials of 1432-avoiding permutations.","Restricted to 321-avoiding permutations, the same formula agrees with the known flagged set-valued Young tableau formula for that family, so the Rothe-tableau model contains the older model as a special case.","The converse direction is an obstruction: if $w$ contains a 1432 pattern, the double Grothendieck polynomial cannot be represented by this particular Rothe-tableau sum, and the failure is visible already at the level of single Schubert polynomials.","Via tableau complexes, two equivalent formulas hold for 1432-avoiding permutations, one using limit set-valued Rothe tableaux and one using single-valued Rothe tableaux with correction factors."],"supporting_citations":[{"why":"Defines the double Grothendieck polynomials by isobaric divided differences, the object the formula is about.","marker":"[21]"},{"why":"Gives the earlier flagged set-valued Young tableau formula for 2143-avoiding permutations, the model this paper generalizes.","marker":"[18]"},{"why":"Introduces set-valued Young tableaux and the set order on finite subsets used in the row and column conditions.","marker":"[7]"},{"why":"Supplies the 321-avoiding tableau formula recovered in Corollary 2.3 and the operator identity used as Lemma 2.12 in the induction.","marker":"[24]"},{"why":"Provides the three K-polynomial expressions for tableau complexes used to prove the two alternative formulas.","marker":"[17]"},{"why":"Introduced the induction through symmetrizing operators for tableau formulas, the proof strategy adapted here.","marker":"[30]"},{"why":"Extended that strategy to Grothendieck polynomials on Young diagrams, the technique here transplanted to Rothe diagrams.","marker":"[25]"},{"why":"Identifies the Rothe diagram of a 321-avoiding permutation with a reflected skew Young diagram, the identification behind the specialization.","marker":"[6]"},{"why":"Gives the balanced-labeling model of Schubert polynomials used to prove the failure for 1432-containing permutations.","marker":"[11]"}],"fun_headline_variants":["Set-valued Rothe tableaux characterize 1432-avoiding permutations","Double Grothendieck polynomials via set-valued Rothe for 1432-avoiders","Grothendieck polynomials from set-valued Rothe tableau iff 1432-avoiding","1432-avoiding permutations: set-valued Rothe tableau formula for Grothendieck","Exact set-valued Rothe sum for Grothendieck polynomials: 1432-avoiding"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the equivalence-class case analysis for tableaux of shape $D(w)$ (Theorem 2.10) is complete exactly as sketched, even though the paper gives only a sketch of the proof and relies on the operator identity stated in Lemma 2.12; a missing configuration there would break the induction connecting the two sides.","fun_headline_variants_meta":{"raw":{"variants":["Set-valued Rothe tableaux characterize 1432-avoiding permutations","Double Grothendieck polynomials via set-valued Rothe for 1432-avoiders","Grothendieck polynomials from set-valued Rothe tableau iff 1432-avoiding","1432-avoiding permutations: set-valued Rothe tableau formula for Grothendieck","Exact set-valued Rothe sum for Grothendieck polynomials: 1432-avoiding"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000586,"raw_usage":{"total_tokens":2785,"prompt_tokens":1005,"completion_tokens":1780,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":621,"completion_tokens_details":{"reasoning_tokens":1666}},"tokens_in":621,"tokens_out":1780,"duration_ms":12308,"temperature":1.0,"reasoning_tokens":1666,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:49:22.463915+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Expanding both sides of (1.2) for the 1432-avoiding, non-321-avoiding permutation $w=35142$ and comparing coefficients would test the forward claim: equality on every monomial is exactly what Theorem 2.1 predicts, while any mismatch disproves it. For the reverse claim, the same coefficient comparison for $w=1432$ should exhibit at least one monomial where the two sides differ, since Theorem 2.2 asserts that this obstruction always exists.","supporting_citations":[{"cited_title":"Lascoux and M.-P","cited_arxiv_id":null,"evidence_quote":"Defines the double Grothendieck polynomials by isobaric divided differences, the object the formula is about."},{"cited_title":"Knutson, E","cited_arxiv_id":null,"evidence_quote":"Gives the earlier flagged set-valued Young tableau formula for 2143-avoiding permutations, the model this paper generalizes."},{"cited_title":"Buch, A Littlewood-Richardson rule for the K-theory of Grassmannians, Acta Math","cited_arxiv_id":null,"evidence_quote":"Introduces set-valued Young tableaux and the set order on finite subsets used in the row and column conditions."},{"cited_title":"Matsumura, A tableau formula of double Grothendieck polyno mials for 321- avoiding permutations, Ann","cited_arxiv_id":null,"evidence_quote":"Supplies the 321-avoiding tableau formula recovered in Corollary 2.3 and the operator identity used as Lemma 2.12 in the induction."},{"cited_title":"Knutson, E","cited_arxiv_id":null,"evidence_quote":"Provides the three K-polynomial expressions for tableau complexes used to prove the two alternative formulas."},{"cited_title":"Wachs, Flagged Schur functions, Schubert polynomials, and symmetrizing op- erators, J","cited_arxiv_id":null,"evidence_quote":"Introduced the induction through symmetrizing operators for tableau formulas, the proof strategy adapted here."},{"cited_title":"Matsumura, Flagged Grothendieck polynomials, J","cited_arxiv_id":null,"evidence_quote":"Extended that strategy to Grothendieck polynomials on Young diagrams, the technique here transplanted to Rothe diagrams."},{"cited_title":"Billey, W","cited_arxiv_id":null,"evidence_quote":"Identifies the Rothe diagram of a 321-avoiding permutation with a reflected skew Young diagram, the identification behind the specialization."},{"cited_title":"Fomin, C","cited_arxiv_id":null,"evidence_quote":"Gives the balanced-labeling model of Schubert polynomials used to prove the failure for 1432-containing permutations."}],"review_version":1}