{"id":"e89dbbea-ae3c-4070-a57a-bea6632b1a9b","arxiv_id":"2412.02932","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A new lower bound relates the number of supports of Schubert polynomials to weighted counts of twelve permutation patterns, strengthening previous 132 and 1432 bounds.","lead":"This paper proves a new lower bound on the number of distinct monomial terms in a Schubert polynomial, expressed through how often certain small patterns occur in the indexing permutation. The bound improves earlier results for Schubert and key polynomials and links pattern containment to Newton polytopes.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Theorem 1.3 rests on Algorithm 2 outputting exactly r2(D) diagrams with distinct weights; no lemma proves termination, exact count, or that the pivot-update rule prevents a column pair from being counted twice.","rationale":"This concern is load-bearing because Theorem 1.3 is the engine from which Corollaries 1.2, 1.4, 1.5 and Theorem 1.1 are derived; if Algorithm 2 cannot be certified to produce exactly r2(D) distinct support vectors, none of the advertised lower bounds follow. The reader identified the same spot, and I agree. I do not regard this as evidence of falsity: the algorithm is plausible and the worked example is consistent, but the paper's own exposition omits the necessary invariant and counting lemma. A finite exhaustive check would settle whether the hidden assumption is actually false for small diagrams. I also note that Theorem 4.1's distinctness of pattern-generated subdiagrams is checked by table inspection and Remark 4.2 admits a subtle near-counterexample; that is a second gap, but I would rank Algorithm 2 first because it underpins Theorem 1.3 for all diagrams, not just Rothe diagrams. Therefore I would keep the CONDITIONAL verdict rather than accepting the proof as complete.","tokens_in":15664,"tokens_out":23277,"duration_ms":217498,"concrete_test":"Enumerate all diagrams D in [4]x[4] (2^16 cases) and, for each row pair (i1,i2), implement Steps 0-3 of Algorithm 2 verbatim. Verify (1) every C produced is a well-defined diagram and C= 1+r1+r2+r3 is obtained by counting these diagrams, an overcount or undercount at this step would leave Theorem 1.3 unproved and Theorem 1.1 unsupported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the number theta_w of supports of a Schubert polynomial, equivalently the number of lattice points of its Newton polytope, and proves lower bounds for this number in terms of permutation patterns. The analysis is carried out for dual characters chi_D of flagged Weyl modules associated with arbitrary diagrams D: Theorem 1.3 claims theta_D >= 1 + r1(D) + r2(D) + r3(D), where r1, r2, r3 count occurrences of three local configurations. The proof constructs that many subdiagrams with distinct weight vectors via three algorithms. Specializing D to a Rothe diagram yields Theorem 1.1, a lower bound on theta_w involving twelve pattern classes; specializing to a skyline diagram yields Theorem 1.5 for key polynomials. The paper also gives computational evidence and conjectures on maximal values of theta_w and on positivity of pattern coefficients.","tokens_in":15816,"tokens_out":17576,"duration_ms":183298,"significance":"If the proof is completed, the result is significant: it gives some of the first lower bounds for support counts in terms of permutation patterns, and Corollary 1.2 improves the existing principal specialization bounds of Weigandt and Gao. The flagged Weyl module framework is natural and the bound for all diagrams is considerably more general than a Schubert-polynomial-specific statement. The proof strategy is constructive and parameter-free, and Remark 4.2 honestly notes that the pattern-to-subdiagram map is not surjective. However, the main theorem currently rests on several unproved invariants of Algorithm 2 and on incompletely justified distinctness arguments, so the central claim is not yet established as written.","major_comments":[{"comment":"The proof of Theorem 1.3 requires that Algorithm 2, for each fixed row pair (i1, i2), outputs exactly the sum over j1 < j2 of r2(D; i1, i2; j1, j2) diagrams, that every output is a diagram C < D, and that the output weights are distinct both within S_{i1,i2}(D) and across different row pairs. No lemma states or proves termination, the exact output count, or the invariant that makes Step 3's pivot-update rule prevent a column pair from being counted once when the smaller column is the pivot and again when the larger column is the pivot. The statement after Step 0 that the cells (i1 - m, j1) and (i1 - m, j2) are blank for every m <= r2 is asserted without proof; if that invariant fails, Step 2 can try to move a box into an occupied cell or produce an object that is not <= D. Propositions 3.3 and 3.4 presuppose these properties rather than prove them, so the count r1 + r2 + r3 that yields the lower bound (1.4) is not established.","section":"Section 3.2, Algorithm 2 and Eq. (1.4)"},{"comment":"The distinctness proof for the diagrams produced by Algorithm 2 contains incorrect counting. In Case 2 the paper says that if C is generated before C' and the two correspond to distinct pivots, then C has fewer boxes than C' in row i1; the opposite is true, because a later diagram has the earlier pivot boxes removed from row i1. In Case 1' the number of boxes of D in row i1 moved by Step 0 for the pair (i1', i2') is asserted to be exactly k = #{j : r1(D; i1, j) > 0}, but Step 0 moves every box above row i1', including boxes in row i1 with no blank box above them, so the asserted equality is not generally valid. These claims are load-bearing for the mutual distinctness of the diagrams in S_{i1,i2}(D) and for their distinctness from the diagrams produced by the other algorithms.","section":"Proposition 3.3, Case 2 and Case 1'"},{"comment":"The argument that a diagram from Algorithm 2 and a diagram from Algorithm 1 cannot have equal weights is not valid as written. From the fact that C has fewer boxes than D in row i2, the paper infers that some boxes of D in row i2 must be moved up to form C' and then that 'the boxes of D lying above row i2 are moved up to the topmost positions' in Algorithm 1. Equal total weight only forces some compensating shift somewhere; it does not imply that Algorithm 1 compacts all rows above i2. The subsequent assertion that k boxes of row i1 are moved in Algorithm 1 repeats the counting problem noted in the previous comment. Since this proposition is needed to separate the r1 and r2 contributions, it requires a correct proof.","section":"Proposition 3.4"},{"comment":"The distinctness of the subdiagrams in Sub(D(w)) is the step that turns pattern counts into a lower bound for r1 + r2 + r3, but it is justified only by the informal claim that a subdiagram determines its generating pattern. The proof of 'D1_sub = D2_sub implies they are generated by the same u pattern' assumes the uniqueness it is meant to prove, and no case analysis is given for the twelve pattern types in Tables 4 and 5. In addition, the text does not discuss the possible overlap when w^{-1}(j) equals one of the listed row indices, which would reduce the recovered position set from five elements to fewer. This is a finite verification and may be routine, but as written the injectivity assertion is not demonstrated; for u = 15342, the claim that the two subdiagrams are 'obviously distinct' is not a proof.","section":"Section 4.1, Theorem 4.1"}],"minor_comments":[{"comment":"There are several typographical errors: 'Schube rt' in the abstract, 'Gvien' for 'Given', 'nubmer' for 'number', 'expect for n = 7' for 'except for n = 7', and 'Stanely' in the reference to Conjecture 5.4.","section":"Throughout"},{"comment":"In the proof of Theorem 4.3 the notation D(w) is used where D(alpha) is intended; for example equation (4.3) should read r2(D(alpha)), not r2(D(w)).","section":"Section 4.2, Theorem 4.3"},{"comment":"The symbol D is used ambiguously in Algorithm 2: r2(D; i1, i2; j, j') refers to the original input diagram, while the conditions in Steps 1 and 2 are evaluated on the current diagram after Step 0 and after subsequent pivot moves. The algorithm should use a separate symbol for the current diagram and should state explicitly which diagram each r2 expression refers to.","section":"Algorithm 2"},{"comment":"The proof of Proposition 3.2 cites [17, Lemma 18] for the fact that C < D implies that C and D have distinct weight vectors. A one-line proof would make the section more self-contained and would avoid importing a result from the reference.","section":"Proposition 3.2"},{"comment":"The assertion about the blank cells (i1 - m, j1) and (i1 - m, j2) should mention the elementary bound m <= r2(D; i1, i2; j1, j2) <= i1 - 1, so that the indices i1 - m remain valid row indices.","section":"Step 0 of Algorithm 2"}],"recommendation":"major_revision","confidential_remarks":"The main theorem may well be true; the problems identified are missing algorithm invariants and incomplete distinctness arguments rather than an explicit counterexample. The proof of Theorem 4.1 is a finite verification and should be made fully explicit, possibly with a machine-checked table. I see no circularity: reference [9] is used only for background on an equality criterion and is not used to prove the main bounds. The paper fits the scope of the journal, but the load-bearing gaps in Section 3 and Section 4.1 need to be repaired before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper gives a real strengthening of known lower bounds for theta_D—the number of supports of dual flagged Weyl characters—and hence for Schubert polynomials. Theorem 1.3, theta_D >= 1 + r1 + r2 + r3, is a natural and meaningful extension of the Meszaros–St.Dizier–Tanjaya bound, and the specialization to skyline diagrams (Theorem 4.3) is exact and elegant. Theorem 1.1 improves on the 132/1432 bounds of Weigandt and Gao by bringing in a list of S5 patterns. The strategy is honest: construct subdiagrams directly from diagrams/patterns, no fitted constants, no circularity. The one self-citation is background, not load-bearing.\n\nThe soft spots are real but fixable. Most importantly, Section 3.2 never proves that Algorithm 2 outputs exactly r2(D) diagrams. It states that running over all row pairs gives the desired number, and it proves distinctness of the weight vectors, but there is no lemma showing that for fixed (i1,i2) the set S_{i1,i2}(D) has size equal to the sum defining r2 for that pair. The stress-test concern about double-counting a column pair from both pivots is legitimate; the 'move the pivot up' update and the weight-distinctness arguments appear to handle it, but only implicitly. Termination is also asserted rather than proven. Similarly, the injectivity check in Theorem 4.1—recovering the pattern from the subdiagram—is done by table inspection plus a brief argument; plausible, but a systematic case analysis would be more persuasive. Minor typos aside, these are gaps in exposition and proof details, not signs of a wrong result.\n\nIf the missing count lemma can be supplied (and I suspect it can), the paper is a solid contribution. The bounds themselves are likely correct and the skyline formula is a nice clean statement. The examples and tables help, but the core proof needs tightening before it goes into a journal.\n\nI would send this to peer review and would bring it to reading group. A good referee can request the missing lemma and a more formal distinctness verification without much risk of hitting a fatal flaw.","headline":"Genuine advance on Schubert support-count bounds; the constructive proof is sound in strategy but needs a missing count lemma for Algorithm 2.","tokens_in":16450,"tokens_out":3951,"would_cite":true,"duration_ms":47273,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E10","05E14","05A19","14N15"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every permutation, the paper proves that the number of supports of its Schubert polynomial is at least one plus a weighted count of twelve patterns.","keywords":["Schubert polynomial","support","Newton polytope","flagged Weyl module","dual character","pattern containment","principal specialization","key polynomial"],"falsifier":"Choose a diagram small enough for direct expansion, such as the $6\\times 6$ example in Section 3, compute $\\chi_D$ explicitly, and compare the number of distinct exponent vectors with $1+r_1(D)+r_2(D)+r_3(D)$; a single diagram where the latter exceeds the former would refute Theorem 1.3. Alternatively, run Algorithm 2 on each row pair of that diagram and count the output: a pair $(i_1,i_2)$ yielding fewer than $r_2(D;i_1,i_2)$ diagrams would isolate exactly where the missing lemma would have to fail.","tokens_in":15376,"feed_emoji":"🧩","tokens_out":10991,"duration_ms":93773,"temperature":0.7,"pith_summary":"The paper establishes a lower bound on how many distinct monomials a Schubert polynomial can have. For every permutation $w$, the number $\\theta_w$ of monomials (equivalently, the number of lattice points in the Newton polytope of the Schubert polynomial) is at least $1$ plus a weighted count of occurrences of twelve specific permutation patterns, with weights ranging from $1$ to $4$. The real engine is a general theorem for all diagrams: for the dual character $\\chi_D$ of the flagged Weyl module of a diagram $D$, $\\theta_D \\geq 1 + r_1(D) + r_2(D) + r_3(D)$, where the $r_i$ count four-, six-, and eight-box configurations in $D$. Since every Schubert polynomial is such a character for its Rothe diagram, the pattern bound follows; since $\\theta_w$ is always at most the principal specialization $\\nu_w$, the same inequality immediately improves the previously known lower bounds for $\\nu_w$. For key polynomials, the general bound translates into a sum over inversion triples and quadruples of the composition.","feed_headline":"Patterns force extra Schubert monomials","feed_subtitle":"A new lower bound shows Schubert polynomials carry at least one plus pattern-weighted monomials.","key_machinery":"The load-bearing machinery is the support description of dual flagged Weyl characters: for a diagram $D=(D_1,\\ldots,D_n)$, the supports of $\\chi_D$ are exactly the weight vectors $\\mathrm{wt}(C)$ of diagrams $C$ with $C_j \\leq D_j$ in Gale order for every column. The argument then builds three explicit algorithms that manufacture such diagrams below $D$ from the counted configurations. Algorithm 1 moves an occupied box up into a blank box directly above it, realizing each $r_1$ configuration one at a time. Algorithm 2 fixes a row pair $(i_1,i_2)$, compactifies the boxes above $i_1$ to the top of their columns, then lifts selected boxes of row $i_2$ into the blank rows $i_1-1,i_1-2,\\ldots$ to realize the $r_2$ configurations. Algorithm 3 fixes a pivot box $(i,j)$, applies Algorithm 1 to the subdiagram below row $i$, and raises the pivot into the blank space above it, realizing the $r_3$ configurations. The distinctness assertions in Propositions 3.2–3.8 are what turn this construction into a counting lower bound.","core_discovery":"The central claim is Theorem 1.3: for every diagram $D$ in $[n]\\times[n]$, $\\theta_D \\geq 1+r_1(D)+r_2(D)+r_3(D)$. Here $r_1$ counts subdiagrams equal to a two-box configuration (one occupied box below one blank box in the same column), $r_2$ counts six-box configurations of types (B) and (B$'$), and $r_3$ counts four-box configurations of types (C), (C$'$), and (C$''$). When $D$ is the Rothe diagram of $w$, this becomes Theorem 1.1, the twelve-pattern lower bound for $\\theta_w$; when $D$ is the skyline diagram of $\\alpha$, it becomes Theorem 1.5, a lower bound for the support count of the key polynomial. The proof constructs, for each counted configuration, a diagram $C$ that is $\\leq D$ in Gale order; each such $C$ has a weight vector, and Proposition 2.1 identifies the set of weight vectors of diagrams below $D$ with the supports of $\\chi_D$. Propositions 3.2–3.8 show that the combined output of the three algorithms has pairwise distinct weight vectors, so the total output injects into the support set.","pith_inferences":["If the missing output-count lemma for Algorithm 2 is supplied, the same three-algorithm template could be applied to new configuration families, potentially pushing the right-hand side of (1.1) toward the empirical $d_u$ values in Table 3, where already $d_u > c_u$ for $u=13452786$.","Because Theorem 1.3 is stated for arbitrary diagrams rather than permutations, it may transfer to other diagram-indexed polynomial families beyond Schubert and key polynomials, provided the analogue of Proposition 2.1 holds; a concrete test is the double Schubert polynomial computed from a Rothe diagram.","The near-coincidence of maximizers for $\\alpha_n$ and $\\beta_n$ in Table 2, with a mismatch at $n=7$, suggests the layered-permutation conjectures for $\\nu$ and $\\theta$ are not equivalent; proving Conjecture 5.2 would require a bound sharp for layered permutations, which Theorem 1.1 is not obviously."],"forward_implications":["For every permutation $w$, $\\nu_w=\\mathfrak{S}_w(1,\\ldots,1)$ is at least the right-hand side of (1.1), improving the earlier pattern bounds $\\nu_w \\geq 1+p_{132}(w)$ and $\\nu_w \\geq 1+p_{132}(w)+p_{1432}(w)$.","For every weak composition $\\alpha$, both $\\theta_{D(\\alpha)}$ and $\\kappa_\\alpha(1,\\ldots,1)$ are bounded below by $1 + \\sum_{(i_1,i_2)\\in\\mathrm{inv}_1(\\alpha)}(\\alpha_{i_2}-\\alpha_{i_1}) + \\sum_{(i_1,i_2,i_3)\\in\\mathrm{inv}_2(\\alpha)}(\\alpha_{i_2}-\\alpha_{i_3})(\\alpha_{i_3}-\\alpha_{i_1}) + \\sum_{(i_1,i_2,i_3,i_4)\\in\\mathrm{inv}_3(\\alpha)}(\\alpha_{i_2}-\\alpha_{i_1})(\\alpha_{i_4}-\\alpha_{i_3})$, ","For a zero-one Schubert polynomial, where every coefficient is $0$ or $1$, the support count equals the principal specialization, so the pattern lower bound applies to the specialization itself.","The computer data in Section 5 suggest that the maximal support count over $S_n$ is attained by layered permutations for $n \\le 9$ and that the pattern-expansion coefficients $d_u$ of $\\theta_w$ are nonnegative for $n \\le 8$; the paper leaves both as conjectures."],"supporting_citations":[{"why":"Establishes that supports of Schubert polynomials are in bijection with lattice points of their Newton polytopes, so $\\theta_w$ and $\\theta_D$ are lattice-point counts.","marker":"[5]"},{"why":"Introduces the rank statistic $r_1(D)$ and proves $\\chi_D(1,\\ldots,1) \\geq 1+r_1(D)$; its lemma technique is the seed of Algorithm 1 and the baseline this paper strengthens.","marker":"[17]"},{"why":"Proves the principal-specialization bound $\\nu_w \\geq 1+p_{132}(w)+p_{1432}(w)$ that Corollary 1.2 improves.","marker":"[8]"},{"why":"Proves $\\nu_w \\geq 1+p_{132}(w)$ and resolves the 132-pattern conjecture for principal specializations; the first pattern lower bound.","marker":"[22]"},{"why":"Identifies the dual character of the flagged Weyl module of a Rothe diagram with the Schubert polynomial, carrying the general theorem to permutations.","marker":"[11, 12]"},{"why":"Identifies the dual character of the flagged Weyl module of a skyline diagram with the key polynomial, carrying the general theorem to compositions.","marker":"[3]"}],"fun_headline_variants":["Patterns force extra Schubert monomials","Patterns guarantee more Schubert terms","Pattern counts boost Schubert supports","New bound: patterns add monomials","Schubert monomials: pattern-powered boost"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on the unproved guarantee that Algorithm 2, for every row pair $(i_1,i_2)$, produces exactly $r_2(D;i_1,i_2)$ distinct diagrams below $D$, with weight vectors distinct from those of Algorithms 1 and 3; if that count ever falls short, the inequality $\\theta_D \\geq 1 + r_1(D)+r_2(D)+r_3(D)$ does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Patterns force extra Schubert monomials","Patterns guarantee more Schubert terms","Pattern counts boost Schubert supports","New bound: patterns add monomials","Schubert monomials: pattern-powered boost"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00017,"raw_usage":{"total_tokens":1261,"prompt_tokens":934,"completion_tokens":327,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":550,"completion_tokens_details":{"reasoning_tokens":261}},"tokens_in":550,"tokens_out":327,"duration_ms":4114,"temperature":1.0,"reasoning_tokens":261,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T22:57:02.343101+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Choose a diagram small enough for direct expansion, such as the $6\\times 6$ example in Section 3, compute $\\chi_D$ explicitly, and compare the number of distinct exponent vectors with $1+r_1(D)+r_2(D)+r_3(D)$; a single diagram where the latter exceeds the former would refute Theorem 1.3. Alternatively, run Algorithm 2 on each row pair of that diagram and count the output: a pair $(i_1,i_2)$ yielding fewer than $r_2(D;i_1,i_2)$ diagrams would isolate exactly where the missing lemma would have to fail.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes that supports of Schubert polynomials are in bijection with lattice points of their Newton polytopes, so $\\theta_w$ and $\\theta_D$ are lattice-point counts."},{"cited_title":"M´ esz´ aros, A","cited_arxiv_id":null,"evidence_quote":"Introduces the rank statistic $r_1(D)$ and proves $\\chi_D(1,\\ldots,1) \\geq 1+r_1(D)$; its lemma technique is the seed of Algorithm 1 and the baseline this paper strengthens."},{"cited_title":"Gao, Principal specializations of Schubert polynomials and patt ern containment, European J","cited_arxiv_id":null,"evidence_quote":"Proves the principal-specialization bound $\\nu_w \\geq 1+p_{132}(w)+p_{1432}(w)$ that Corollary 1.2 improves."},{"cited_title":"Weigandt, Schubert polynomials, 132-patterns, and Stanle y’s conjecture, Algebr","cited_arxiv_id":null,"evidence_quote":"Proves $\\nu_w \\geq 1+p_{132}(w)$ and resolves the 132-pattern conjecture for principal specializations; the first pattern lower bound."},{"cited_title":"Demazure, Une nouvelle formule des caract´ eres, Bull","cited_arxiv_id":null,"evidence_quote":"Identifies the dual character of the flagged Weyl module of a skyline diagram with the key polynomial, carrying the general theorem to compositions."}],"review_version":1}