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Schubert polynomials and patterns in permutations

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Pith's one-line read 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.

desk verdict Genuine advance on Schubert support-count bounds; the constructive proof is sound in strategy but needs a missing count lemma for Algorithm 2. read the letter →

arxiv 2412.02932 v1 pith:QCO322HY submitted 2024-12-04 math.CO math.AGmath.RT

Peter L. Guo , Zhuowei Lin This is my paper · ORCID
classification math.COmath.AGmath.RT MSC 05E1005E1405A1914N15
keywords SchubertpolynomialsupportNewtonpolytopeflaggedWeylmoduledualcharacterpatterncontainmentprincipalspecializationkey
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

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.

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 (4)
  1. [Section 3.2, Algorithm 2 and Eq. (1.4)] 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.
  2. [Proposition 3.3, Case 2 and Case 1'] 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.
  3. [Proposition 3.4] 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.
  4. [Section 4.1, Theorem 4.1] 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.
minor comments (5)
  1. [Throughout] 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.
  2. [Section 4.2, Theorem 4.3] 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)).
  3. [Algorithm 2] 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.
  4. [Proposition 3.2] 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.
  5. [Step 0 of Algorithm 2] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the lower bounds are proved by explicit diagram constructions, with pattern occurrences mapped injectively to subdiagrams; no fitted constants or self-referential definitions are load-bearing.

full rationale

The paper's central claims, Theorems 1.1 and 1.3, are derived by direct combinatorial construction rather than by definitional equivalence. Theorem 1.3 defines r1(D), r2(D), r3(D) as counts of prescribed local configurations in a diagram D, while theta_D is independently defined as the number of supports of the dual flagged Weyl character, equivalently the number of distinct weight vectors of diagrams C <= D by Proposition 2.1. The proof then uses Algorithms 1, 2, and 3 to exhibit r1 + r2 + r3 diagrams with distinct weight vectors; the distinctness arguments in Propositions 3.2, 3.3, 3.6, 3.7, and 3.8 compare row counts and do not presuppose the desired lower bound. The only external lemma cited in this construction is [17, Lemmas 17 and 18], which is not authored by the present authors. The self-citation [9] appears only in the introduction as background on a zero-one criterion and is not used in the proof of the main theorems, so it is not load-bearing. Theorem 4.1 then associates to each occurrence of the relevant patterns in w a subdiagram of the Rothe diagram D(w) via Tables 1, 4, and 5, and proves injectivity by explicitly recovering the unique pattern from each subdiagram; this is an injectivity check rather than a renaming of known results. The skeptic's concern that Algorithm 2 may not be proven to output exactly r2(D) diagrams, or may double-count, is a potential correctness gap in the proof, but it is not circularity: the claimed bound is not being assumed as an input, fitted to data, or derived from the same result. No fitted parameters, renormalization choices, or imported uniqueness theorems force the conclusion.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters or invented entities appear; the coefficients in the pattern bound are combinatorial counts, not fitted numbers. The main external inputs are standard facts about flagged Weyl modules and Gale order.

assumptions (4)
  • domain assumption The set of supports of chi_D(x) is {wt(C) : C <= D}.
    Used as Proposition 2.1 to translate the problem into counting diagrams C <= D with distinct weight vectors.
  • standard math If C < D in the Gale order, then C and D have distinct weight vectors.
    Known lemma from [17], used in Propositions 3.2 and elsewhere to conclude that each constructed diagram has a new weight vector.
  • domain assumption chi_{D(w)}(x) = S_w(x) and chi_{D(alpha)}(x) = kappa_alpha(x).
    Known specializations of dual flagged Weyl characters to Schubert and key polynomials; they let Theorems 1.1 and 1.5 follow from Theorem 1.3.
  • domain assumption The number of supports equals the number of lattice points in the Newton polytope.
    Standard result from [5], used to identify theta_w with the lattice point count in the paper's framing.

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Cite this review

Pith. "Pith review of Schubert polynomials and patterns in permutations." pith.science (2026). https://pith.science/paper/QCO322HY

@misc{pith2026241202932,
  author       = {Pith},
  title        = {Pith review of: Schubert polynomials and patterns in permutations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QCO322HY}},
  note         = {Machine review of arXiv:2412.02932}
}
abstract

This paper investigates the number of supports of the Schubert polynomial $\mathfrak{S}_w(x)$ indexed by a permutation $w$. This number also equals the number of lattice points in the Newton polytope of $\mathfrak{S}_w(x)$. We establish a lower bound for this number in terms of the occurrences of patterns in $w$. The analysis is carried out in the general framework of dual characters of flagged Weyl modules. Our result considerably improves the bounds for principal specializations of Schubert polynomials or dual flagged Weyl characters previously obtained by Weigandt, Gao, and M{\'e}sz{\'a}ros--St. Dizier--Tanjaya. Some problems and conjectures are discussed.

Figures

Figures reproduced from arXiv: 2412.02932 by the authors.

Figure 1.1
Figure 1.1. Configurations for Theorem 1.3. matrix coordinate. Define • r1(D): the number of subdiagrams of D which are equal to the configuration (A) in [PITH_FULL_IMAGE:figures/full_fig_p003_1_1.png] view at source ↗
Figure 2.2
Figure 2.2. A diagram in [4] × [4]. Let R = {r1 < · · · < rk} and S = {s1 < · · · < sk} be two k-element subsets of [n]. We say R ≤ S if ri ≤ si for 1 ≤ i ≤ k. This defines a partial order on all k-element subsets of [n], which is usually called the Gale order. For two diagrams C = (C1, . . . , Cn) and D = (D1, . . . , Dn), denote C ≤ D if Cj ≤ Dj for each 1 ≤ j ≤ n. Let GL(n, C) be the general linear group of n × n invertible … view at source ↗
Figure 2.3
Figure 2.3. The Rothe diagram of w = 1432 and the skyline diagram of α = (1, 3, 0, 2). χD(w)(x) = Sw(x) [11, 12] and χD(α)(x) = κα(x) [3]. 3 Proof of Theorem 1.3 Let D be a given diagram in [n]×[n]. For simplicity, we denote ri = ri(D) for i = 1, 2, 3. To finish the proof of Theorem 1.3, by Proposition 2.1, it suffices to construct r1 +r2 +r3 diagrams less than D which have distinct weight vectors. To accomplish this, we shall … view at source ↗
Figures from the paper (9 more)
Figure 3.4
Figure 3.4. Figure 3.4: An illustration for performing Algorithm 1. [PITH_FULL_IMAGE:figures/full_fig_p007_3_4.png]
Figure 3.5
Figure 3.5. Figure 3.5: A diagram D in [6] × [6]. Algorithm 2. The algorithm will be performed for any given pair of row indices 1 < i1 < i2 ≤ n. Running over all row index pairs, we receive the desired r2 diagrams less the D with distinct weights. Step 0: Let us fix 1 < i1 < i2 ≤ n. Set Si…
Figure 3.6
Figure 3.6. Figure 3.6: An illustration for Step 0 in Algorithm 2. [PITH_FULL_IMAGE:figures/full_fig_p008_3_6.png]
Figure 3.7
Figure 3.7. Figure 3.7: An illustration for Step 2 in Algorithm 2. [PITH_FULL_IMAGE:figures/full_fig_p009_3_7.png]
Figure 3.8
Figure 3.8. Figure 3.8: The diagram by moving the pivot up by a unit. [PITH_FULL_IMAGE:figures/full_fig_p009_3_8.png]
Figure 3.9
Figure 3.9. Figure 3.9: The first two diagrams generated in the second round of [PITH_FULL_IMAGE:figures/full_fig_p010_3_9.png]
Figure 3.10
Figure 3.10. Figure 3.10: The second two diagrams generated in the second roun [PITH_FULL_IMAGE:figures/full_fig_p010_3_10.png]
Figure 3.11
Figure 3.11. Figure 3.11: An illustration for Step 0 in Algorithm 3. [PITH_FULL_IMAGE:figures/full_fig_p012_3_11.png]
Figure 4.12
Figure 4.12. Figure 4.12: A subdiagram of D(162435). cannot be generated by any u ∈ S5 pattern of w. If it were generated by a u pattern, then the u pattern of w would be 16243 and so u = 15243. However, in view of the construction in [PITH_FULL_IMAGE:figures/full_fig_p016_4_12.png]

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