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Signed puzzles for Schubert coefficients

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Pith's one-line read The paper proves a signed puzzle rule that computes every Schubert coefficient, and uses it to prove that fixed-inversion coefficient sums are polynomial.

desk verdict A plausible and likely correct signed puzzle rule for Schubert coefficients, but the proof of the central row-by-row correspondence is under-specified and needs either a filled-in proof or a computational check. read the letter →

arxiv 2504.17734 v2 pith:TBYYLCCD submitted 2025-04-24 math.CO

Igor Pak , Colleen Robichaux This is my paper · ORCID
classification math.CO MSC 14N1505B45
keywords SchubertcoefficientssignedpuzzleruleKnutsonrecurrenceBruhatorderEhrharttheorypolynomialityflagvarietycohomology
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

Schubert coefficients are the integers that record products in the cohomology of flag varieties, and finding combinatorial formulas for them is a long-standing open problem. This paper proves that every Schubert coefficient $c^w_{u,v}$ satisfying the dimension equation $\operatorname{inv}(u)+\operatorname{inv}(v)=\operatorname{inv}(w)$ equals the signed count of tilings of a finite parallelogram by labeled triangles. The construction is the first signed puzzle rule that works for all Schubert coefficients, rather than for special families. The rule is built directly from a recurrence: one row of the tiling encodes one step of the recurrence, so iterating over $\binom n2-\operatorname{inv}(u)$ rows reproduces the coefficient. As an application, the paper proves that the sum $\gamma_k(n)$ of all coefficients with $\operatorname{inv}(w)=k$ is a polynomial in $n$, a structural conclusion that follows from an Ehrhart-theoretic count of the puzzles.

What carries the argument

The central object is the signed puzzle itself: a parallelogram region tiled by $O(n^9)$ types of unit equilateral triangles, colored white, shaded, and dark (the dark divided into yellow, blue, and red), carrying three layers of labels—permutation labels $(a,b,c)$, feedback labels with blank entries, and transmuter labels $(g,h)$ with $g<h$—together with edge indicators $\circ$ and $\ast$ that force exactly one dark triangle per row. The load-bearing mechanism is the row-by-row correspondence with Knutson's recurrence: in each row the unique dark triangle marks the index $i$ of the recurrence, and the labels record how the three permutations change. The overall sign of a puzzle is the parity of the number of red dark triangles, which matches the signs of the positive and negative terms in the final case of the recurrence.

What would settle it

Enumerate by brute force all tilings of the region $\Gamma$ for a small triple with a known coefficient (for example $n=4$, $u=2143$, $v=3412$, $w=4321$) and compare the signed sum to $c^w_{u,v}$ computed from Schubert polynomials; any mismatch falsifies the theorem.

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Extended reading notes

Core claim

On its own terms, the central discovery is a complete signed puzzle rule for Schubert coefficients. Given permutations $u,v,w\in S_n$ satisfying the dimension equation, one draws an $n\times \ell$ parallelogram $\Gamma$ whose top boundary carries the triples $(u(i),v(i),w(i))$ and whose bottom boundary carries the reversed triples $(n-i+1,i,n-i+1)$; a puzzle is a tiling by the finite set $T_n$ of white, shaded, and dark labeled unit triangles, and the sign of a tiling is $(-1)^p$, where $p$ is the number of red triangles in it. The paper proves that summing these signs over all tilings returns $c^w_{u,v}$. The proof works row by row: each row of the tiling realizes exactly one of the cases of Knutson's recurrence, the unique dark triangle in the row selects the index $i$ where the recurrence is applied, and the feedback and transmuter labels implement the transpositions that appear in the recurrence, so the whole tiling is a geometric trace of the recurrence expansion.

Load-bearing premise

The construction assumes that every valid row of tiles corresponds faithfully to one step of the recurrence, including the non-local transpositions implemented by transmuter labels, and that this translation has no exceptional edge case.

Editorial extensions

If this is right

  • For every triple satisfying the dimension equation, $c^w_{u,v}$ equals the signed count of a finite tiling, so Schubert coefficients have an explicit GapP-type formula with a piece set of size $O(n^9)$.
  • For each fixed $k$, the total $\gamma_k(n)$ is a polynomial in $n$; the proof gives degree $O(k^2)$, which is weaker than the elementary $6k$ bound, so the polynomiality itself is the new content.
  • A puzzle for $c^w_{u,v}$ has exactly $\binom n2-\operatorname{inv}(u)$ rows, so the signed count is a linear-depth trace of the recurrence rather than a sum over independent cases.
  • The modified piece set $T'_n$ described in Remark 5.1, which enforces the non-local constraint through an added label inequality, also computes the same coefficients and is the version needed for the polynomiality argument.

Reading between the lines

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

  • For special families of permutations, the signed contributions may cancel in pairs, so the same construction could potentially be refined to a manifestly positive puzzle rule; the paper does not attempt this.
  • The architecture of encoding a recurrence as labeled tile rows and counting fixed-height regions by Ehrhart theory should transfer to equivariant or K-theoretic Schubert coefficients if a suitable recurrence exists; the paper notes this possibility without proving it.
  • For small $n$, the signed puzzle count can serve as a computational certificate against independent pipe-dream or Schubert polynomial calculations, giving a quick consistency check for any proposed simplification of the piece set.
  • The $O(n^9)$ piece count is a proof-of-concept rather than an efficient algorithm; allowing rotations or quotienting by symmetries would be a natural next step, but the paper uses only parallel translations.
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Editorial analysis

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

Referee Report

4 major / 6 minor

Summary. The paper introduces a signed puzzle rule for Schubert coefficients: for every n and every triple u,v,w in S_n satisfying inv(u)+inv(v)=inv(w), the signed count of tilings of an n-by-(binom(n,2)-inv(u)) parallelogram region by a set T_n of O(n^9) labeled triangular pieces is claimed to equal the Schubert coefficient c^w_{u,v}. The proof is by induction on the number of rows, reducing to a row-by-row simulation of Knutson's recurrence. As an application, the paper proves that the sum gamma_k(n) of Schubert coefficients over permutations w with inv(w)=k is a polynomial in n, using an Ehrhart-theoretic argument on unions of rational polyhedra associated to relative placements of the special tiles.

Significance. If correct, this is the first signed puzzle rule that applies to all Schubert coefficients, extending earlier special cases and providing a new type of combinatorial rule in an area where even signed interpretations are scarce. The polynomiality theorem for gamma_k(n) is a genuinely new structural result, and the proof strategy connecting tile placements to Ehrhart theory of unimodular polyhedra is interesting in its own right. The construction is explicit, richly illustrated, and the authors are transparent about the role of Knutson's recurrence. The main weakness is that the central verification in Section 5 is incomplete at exactly the point where non-local transpositions are implemented; without that verification, the equality in Theorem 1.1 is not established.

major comments (4)
  1. [Section 5, paragraphs on dark blue and dark red triangles] The proof of Theorem 1.1 rests on the claim that each puzzle row simulates one application of Lemma 3.1, but the implementation of the non-local transpositions t_{jk} in case (3) via transmuter labels is not verified. The text states that the translation is 'immediate' and that 'The details are straightforward', yet it does not specify how the transmuter labels determine the index k, how the condition |{j,k}∩{i,i+1}|=1 is enforced, or why the bottom permutation labels are exactly u t_{jk}. This is the load-bearing step of the entire paper. Please provide a complete, case-by-case proof of the transmuter mechanism, or supply a machine-checked enumeration for small n and k that confirms the signed puzzle count agrees with c^w_{u,v}.
  2. [Section 7.1, duality identity] The identity c^w_{u,v} = c^u_{w,v}, stated with the ambiguous clause 'where w = w·w◦', is used to reduce the parallelogram height to k in the proof of Theorem 1.2, but the precise symmetry and its proof are not given. The notation uses w for two different permutations in the same sentence, making the statement impossible to verify as written. Please restate the duality with distinct variables, prove it or give a precise reference, and then derive the correct summation condition for the new coefficient.
  3. [Section 7.1, displayed equality for gamma_k(n)] The text says 'we will work with a modified set T′n' and then writes 'gamma_k(n) = sum_{puzzle T of Γ with Tn} s(T)', using T_n rather than T′_n. This is either a typo or an unproven assertion. If the intended equality is with T′_n, then Theorem 1.1 must be proved for T′_n, which is not done; Remark 5.1 only sketches the modification. Please correct the displayed equation and clarify whether the equality in Theorem 1.1 holds for the modified tile set.
  4. [Section 7.4, derivation of distance-dependent inequalities] The claim that the condition that the labels in each row form permutations 'gives inequalities relating differences between the labels and distances between the columns' is not derived. In particular, the text asserts inequalities of the form alpha_i - alpha_j > m with m depending on the distance between columns, but a permutation only imposes distinctness of labels, not a lower bound on their difference that grows with column distance. Since the total unimodularity argument in Section 7.5 depends on the inequalities having the form alpha_i - alpha_j <= b, please provide a precise derivation of these inequalities.
minor comments (6)
  1. [Section 2] The notation 'inv(w) := {(i,j): i<j, w(i)>w(j)} denote the number of inversions' confuses the set of inversion pairs with its cardinality; use |inv(w)| or define Inv(w) for the set and inv(w) for its size.
  2. [Section 4.4] In the paragraph on shaded triangles, 'eight of fewer distinct labels' should read 'eight or fewer distinct labels'.
  3. [Section 5, first-ascent argument] The proof that the dark triangle's index i is forced to be the first ascent of u is compressed. It would help to add a short explanation of why no feedback or transmuter labels can affect the shaded triangles to the left of the dark triangle in the first row, so that the permutation labels transmitted there are exactly u(1),...,u(i-1).
  4. [Section 7.1] The phrase 'where w = w·w◦' uses the symbol w for both the original and the transformed permutation; use a different letter such as \tilde{w} for clarity.
  5. [Section 7.2] The terminology 'separated shaded triangles' is introduced after a confusing count of 'at most 2k such shaded triangles, where at most k are not immediately following the dark triangles'. Please clarify exactly which shaded triangles are counted and which are called separated.
  6. [Global] There are several typos: the header of the paper reads 'SCHUBERT T COEFFICIENTS' instead of 'SCHUBERT COEFFICIENTS', and the displayed formula (⊛) has a formatting issue ('the form a,p /∈ [g,h]' should be 'a,p notin [g,h]').

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the signed puzzle rule is a combinatorial encoding of Knutson's external recurrence [Knu03]; no fitted inputs, assumed conclusions, or load-bearing self-citations are present.

full rationale

The paper's central claim, Theorem 1.1, states that a signed puzzle count equals the Schubert coefficient c^w_{u,v}. The proof is an induction on rows of the parallelogram Γ, where the top row of any puzzle is shown to implement one application of Lemma 3.1, Knutson's recurrence from [Knu03]. This is a reduction to an external mathematical theorem, not to the target result or to the authors' own prior work. The puzzle pieces are explicitly constructed so that the dark-triangle colors, dockets, feedback labels, and transmuter labels mirror the four cases of the recurrence; the induction then follows the recurrence path until the terminal coefficient c^{w_0}_{w_0,1}=1. No parameter is fitted to data, no quantity called a prediction is defined from the output it is supposed to predict, and no uniqueness or correctness statement is imported from a self-citation. The discussion in Section 5 about transmuter labels, including the blank-label dark red docket 2 case, is terse and partly delegated to 'the details are straightforward'; that is a verification-gap or correctness concern about the faithfulness of the encoding, not a circularity. Theorem 1.2 is a separate application of Theorem 1.1 together with Ehrhart theory, and it likewise does not assume its conclusion. The only self-references, such as [PR24a] and [PR25], are contextual remarks about other signed interpretations and vanishing results; they are not load-bearing in the proof of Theorem 1.1. Accordingly, the derivation chain is self-contained relative to an independent external recurrence, and no circularity is identified.

Assumptions & free parameters 0 free parameters · 3 assumptions · 1 invented entities

The central claim rests on Knutson's recurrence as an external theorem and on standard Ehrhart theory. The puzzle pieces are introduced as a new construction with no independent evidence outside the proof. No numerical parameters are fitted to data.

assumptions (3)
  • domain assumption Knutson's recurrence (Lemma 3.1) correctly computes Schubert coefficients in type A.
    The entire puzzle rule is designed to simulate this recurrence; the recurrence is cited from [Knu03] and not proved in this paper.
  • domain assumption Standard properties of Schubert coefficients used in the proof of Theorem 1.2, including the symmetry transformation and Stanley's upper bound.
    These results are cited from [Mac91] and [Sta17] and used in the transformation and bounding of γ_k.
  • standard math Ehrhart quasi-polynomial theory and total unimodularity of matrices with at most one +1 and one -1 per row.
    The proof of polynomiality in Theorem 1.2 relies on standard Ehrhart theory and the fact that such constraint matrices are totally unimodular.
invented entities (1)
  • Puzzle piece set T_n of O(n^9) labeled triangles
    purpose: Defines a signed puzzle count that equals Schubert coefficients for all permutations.
    The piece set is the central new mathematical object; there is no evidence for its validity independent of the proof that it simulates Knutson's recurrence.

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Pith. "Pith review of Signed puzzles for Schubert coefficients." pith.science (2026). https://pith.science/paper/TBYYLCCD

@misc{pith2026250417734,
  author       = {Pith},
  title        = {Pith review of: Signed puzzles for Schubert coefficients},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TBYYLCCD}},
  note         = {Machine review of arXiv:2504.17734}
}
read the original abstract

We give a signed puzzle rule to compute Schubert coefficients. The rule is based on a careful analysis of Knutson's recurrence arXiv:math/0306304. We use the rule to prove polynomiality of the sums of Schubert coefficients with bounded number of inversions.

Figures

Figures reproduced from arXiv: 2504.17734 by the authors.

Figure 4.1
Figure 4.1. Region Γ. We label the boundary of Γ as follows. Label all horizontal edges with triples (a, b, c) ∈ [n] 3 . The top edges are labeled (u(1), v(1), w(1)), . . . , (u(n), v(n), w(n)). The bottom edges are labeled (n, 1, n), (n − 1, 2, n − 1), . . . , (1, n, 1). Left and right edges on the boundary of Γ are marked ◦ and ∗, respectively. We use the term position to mean a particular triangle in Γ [PITH_FULL_IMAGE:figu… view at source ↗
Figure 4.2
Figure 4.2. White, shaded and dark triangles. Three colors of dark triangles: dark yellow, dark blue and dark red. No rotations or reflections of the pieces are allowed, only parallel translations. The sides of the triangles will have labels and indicators, described in Section 4.3. Triangles are allowed to share a side in the puzzle if corresponding side labels and indicators are identical. Finally, in addition to color, all d… view at source ↗
Figure 4.3
Figure 4.3. Possible docket numbers of dark triangles. 4.3. Labels and indicators. Here, labels are numbers and indicators are symbols. The labels and indicators on the triangles will be somewhat involved and defined in stages. Level 0. We place an indicator ◦ or ∗ on the left and right edges of all triangles as follows. For both white and shaded triangles, the indicators on the left and right edges must be equal. For dark tria… view at source ↗
Figures from the paper (13 more)
Figure 4.4
Figure 4.4. Figure 4.4: Indicators ◦ and ∗ on the left and right edges of three types of triangles. a,b,c a,b,c a,b,c a,b,c p,q,r x,y,z a,b,c p,q,r x,y,z [PITH_FULL_IMAGE:figures/full_fig_p005_4_4.png]
Figure 4.5
Figure 4.5. Figure 4.5: Permutation labels on three types of triangles. Level 2. Some triangles have additional triples of feedback labels on the edges. These will be of the form (d, e, f), where d, e, f ∈ ⟨n⟩ and at least one of these is blank. All dark triangles have feedback labels, whic…
Figure 4.6
Figure 4.6. Figure 4.6: Feedback labels on three types of triangles, and a shaded triangle with￾out feedback labels. Level 3. Finally, triangles may have additional transmuter labels on the left and right edges of the form (g, h), where g, h ∈ ⟨n⟩. In white triangles transmuter labels on th…
Figure 4.7
Figure 4.7. Figure 4.7: Transmuter labels on three types of triangles [PITH_FULL_IMAGE:figures/full_fig_p005_4_7.png]
Figure 4.8
Figure 4.8. Figure 4.8: Five types of white triangles. Shaded triangles: There are ten types of labelings for shaded triangles depending on whether they have feedback labels, transmuter labels, or both, see [PITH_FULL_IMAGE:figures/full_fig_p006_4_8.png]
Figure 4.9
Figure 4.9. Figure 4.9: Ten types of shaded triangles. For shaded triangles, there is an additional condition relating indicators and permutation labels: a shaded triangle where the first entries in the permutation labels form an ascent must have ∗ indicators, see [PITH_FULL_IMAGE:figures/…
Figure 4.10
Figure 4.10. Figure 4.10: Indicators of shaded triangles that are allowed under the ascent condition. inequalities for permutation labels correspond precisely to the cases in Knutson’s recursion. This gives O(n 8 ) dark triangles in total. g,h g,h 2 a,b,c p,q,r a,q,c -,b,- -,- b > q c < r 1 …
Figure 4.11
Figure 4.11. Figure 4.11: Label constraints on dark triangles by docket number. 4.5. Summary. We gave a construction of O(n 9 ) puzzle pieces. All pieces are triangles with three types of labels (permutation, feedback and transmuter), two types of indicators (◦ and ∗), five colors (white, sh…
Figure 5.1
Figure 5.1. Figure 5.1: Permutation labels transmitted from top to bottom. Feedback signal dominated the permutation signal resulting in two transpositions: d ↔ a and f ↔ c. The feedback labels in [PITH_FULL_IMAGE:figures/full_fig_p008_5_1.png]
Figure 5.2
Figure 5.2. Figure 5.2: For dark red triangle docket number 2, the blank transmuter labels signal initiates transposition tjk with j = i+ 1 at the following shaded triangle. Here d = u(i), a = u(i + 1) and h = u(k). Finally, we note that some constraints of the recurrence follow from the se…
Figure 6.1
Figure 6.1. Figure 6.1: An example of a puzzle. 7. Proof of Theorem 1.2 7.1. The setup. First, note that c w u,v = c u w,v, where w = w · w◦ . Thus we can rewrite γk(n) = X u,v,w∈Sn : inv(u)=( n 2 )−k c w u,v . Now consider all puzzles of the n × k parallelogram region Γ as in [PITH_FULL_I…
Figure 7.1
Figure 7.1. Figure 7.1: Turning parallelogram region Γ into a rectangle [PITH_FULL_IMAGE:figures/full_fig_p010_7_1.png]
Figure 7.2
Figure 7.2. Figure 7.2: For the example in [PITH_FULL_IMAGE:figures/full_fig_p011_7_2.png]

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Reference graph

Works this paper leans on

33 extracted references · 17 canonical work pages · cited by 2 Pith papers

  1. [1]

    Alexander Barvinok, Lattice points and lattice polytopes, in Handbook of discrete and computational geometry, CRC, Boca Raton, FL, 1997, 133--152

    work page 1997

  2. [2]

    Matthias Beck and Sinai Robins, Computing the continuous discretely (Second ed.), Springer, New York, 2015, 285 pp

    work page 2015

  3. [3]

    284 (2015), 54--111

    Arkady Berenstein and Edward Richmond, Littlewood--Richardson coefficients for reflection groups, Adv.\ Math. 284 (2015), 54--111

    work page 2015

  4. [4]

    Sara Billey, Kostant polynomials and the cohomology ring for G/B , Duke Math. J. 96 (1999), 205–224

    work page 1999

  5. [5]

    Buch, Andrew Kresch, Kevin Purbhoo and Harry Tamvakis, The puzzle conjecture for the cohomology of two-step flag manifolds, J

    Anders S. Buch, Andrew Kresch, Kevin Purbhoo and Harry Tamvakis, The puzzle conjecture for the cohomology of two-step flag manifolds, J. Algebraic Combin. 44 (2016), 973--1007

    2016

  6. [6]

    159 (2005), 407--436

    Haibao Duan, Multiplicative rule of Schubert classes, Invent.\ Math. 159 (2005), 407--436

    work page 2005

  7. [7]

    William Fulton, Young tableaux, Cambridge Univ.\ Press, Cambridge, UK, 1997, 260 pp

    1997

  8. [8]

    Rebecca Goldin and Allen Knutson, Schubert structure operators and K^*_T(G/B) , Pure Appl.\ Math. Q. 17 (2021), 1345--1385

    work page 2021

Show all 33 references
  1. [9]

    Shephard, Tilings and patterns, Freeman, New York, 1987, 700 pp

    Branko Gr\"unbaum and Geoffrey C. Shephard, Tilings and patterns, Freeman, New York, 1987, 700 pp

    1987

  2. [10]

    (2024), no

    Christian Ikenmeyer, Igor Pak and Greta Panova, Positivity of the symmetric group characters is as hard as the polynomial time hierarchy, Int.\ Math.\ Res.\ Not. (2024), no. 10, 8442--8458

    2024

  3. [11]

    10 (2001), 345--353

    Allen Knutson, Descent-cycling in S chubert calculus, Experiment.\ Math. 10 (2001), 345--353

    2001

  4. [12]

    Allen Knutson, A Schubert calculus recurrence from the noncomplex W -action on G/B , preprint (2003), 10 pp.; arXiv:math/0306304

    2003 arXiv

  5. [13]

    Allen Knutson, Schubert calculus and puzzles, in Schubert calculus, MSJ, Tokyo, 2016, 185--209

    2016

  6. [14]

    Allen Knutson, Schubert calculus and quiver varieties, in Proc.\ ICM, Vol. VI, Sec. 12--14, EMS Press, Berlin, 2023, 4582--4605

    2023

  7. [15]

    Allen Knutson and Terence Tao, Puzzles and (equivariant) cohomology of Grassmannians, Duke Math. J. 119 (2003), 221--260

    2003

  8. [16]

    Allen Knutson and Paul Zinn-Justin, Schubert puzzles and integrability I: invariant trilinear forms, preprint (2017), 51 pp.; arXiv:1706.10019

    2017 arXiv

  9. [17]

    Allen Knutson and Paul Zinn-Justin, Schubert puzzles and integrability II: multiplying motivic Segre classes, preprint (2021), 80 pp.; arXiv:2102.00563

    2021 arXiv

  10. [18]

    Allen Knutson and Paul Zinn-Justin, Schubert puzzles and integrability III: separated descents, preprint (2023), 42 pp.; arXiv:2306.13855

    2023 arXiv

  11. [19]

    Macdonald, Notes on Schubert polynomials, Publ.\ LaCIM, UQAM, Montreal, 1991, 116 pp

    Ian G. Macdonald, Notes on Schubert polynomials, Publ.\ LaCIM, UQAM, Montreal, 1991, 116 pp

    1991

  12. [20]

    Laurent Manivel, Symmetric functions, Schubert polynomials and degeneracy loci, SMF/AMS, Providence, RI, 2001, 167 pp

    2001

  13. [21]

    Cecilia Verri, Strip tiling and regular grammars, Theoret.\ Comput.\ Sci

    Donatella Merlini, Renzo Sprugnoli and M. Cecilia Verri, Strip tiling and regular grammars, Theoret.\ Comput.\ Sci. 242 (2000), 109--124

    2000

  14. [22]

    Cristopher Moore, Some polyomino tilings of the plane, preprint (1999), 13 pp.; arXiv:math/9905012

    1999 arXiv

  15. [23]

    Igor Pak, What is a combinatorial interpretation?, in Open Problems in Algebraic Combinatorics, AMS, Providence, RI, 2024, 191--260

    2024

  16. [24]

    Igor Pak and Colleen Robichaux, Signed combinatorial interpretations in algebraic combinatorics, preprint (2024), 24 pp.; to appear in Algebraic Combinatorics; arXiv:2406.13902

    2024 arXiv

  17. [25]

    arXiv:2412.02064 ; extended abstract to appear in Proc.\ 57th STOC (2025), 12 pp

    Igor Pak and Colleen Robichaux, Vanishing of Schubert coefficients (Appendix C joint with David E Speyer), preprint (2024), 30 pp. arXiv:2412.02064 ; extended abstract to appear in Proc.\ 57th STOC (2025), 12 pp

    2024 arXiv

  18. [26]

    Igor Pak and Colleen Robichaux, Vanishing of Schubert coefficients is in assuming the , preprint (2024), 18 pp

    2024

  19. [27]

    153 (2017), 667--677

    Oliver Pechenik and Alexander Yong, Equivariant K -theory of Grassmannians II: the Knutson--Vakil conjecture, Compos.\ Math. 153 (2017), 667--677

    2017

  20. [28]

    Stanley, Chains in the Bruhat order, J.\ Algebraic Combin

    Alexander Postnikov and Richard P. Stanley, Chains in the Bruhat order, J.\ Algebraic Combin. 29 (2009), 133--174

    2009

  21. [29]

    Stanley, Enumerative Combinatorics , vol

    Richard P. Stanley, Enumerative Combinatorics , vol. 1 (Second ed.) and vol. 2, Cambridge Univ. Press, 2012 and 1999, 626 pp.\ and 581 pp

    2012

  22. [30]

    Stanley, Positivity problems and conjectures in algebraic combinatorics, in Mathematics: frontiers and perspectives, AMS, Providence, RI, 2000, 295--319

    Richard P. Stanley, Positivity problems and conjectures in algebraic combinatorics, in Mathematics: frontiers and perspectives, AMS, Providence, RI, 2000, 295--319

    2000

  23. [31]

    Stanley, Some Schubert shenanigans, preprint (2017), 8 pp.; arXiv:1704.00851

    Richard P. Stanley, Some Schubert shenanigans, preprint (2017), 8 pp.; arXiv:1704.00851

    2017 arXiv

  24. [32]

    Peter van Emde Boas, The convenience of tilings, in Complexity, logic, and recursion theory, Dekker, New York, 1997, 331--363

    1997

  25. [33]

    Alexander Yong, Maple 7 code to compute Schubert calculus in G/B , 2006; software available at \\ ayong.web.illinois.edu/papers.html https://ayong.web.illinois.edu/cominrule.v1.0.txt

    2006

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