Pith. sign in

REVIEW 3 major objections 5 minor 2 cited by

Chromatic symmetric functions of claw-free graphs are not Schur positive

T0 review · 3 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Two claw-free graphs disprove a 1995 Schur-positivity conjecture

desk verdict A short, important note refuting Stanley's claw-free Schur positivity conjecture with explicit line graphs; the non-Schur-positive coefficients are computational and need to be made checkable. read the letter →

arxiv 2607.21508 v2 pith:XU7AH4BI submitted 2026-07-23 math.CO

Jacob P. Matherne , Alejandro H. Morales This is my paper · ORCID
classification math.CO MSC 05E0505C1505C31
keywords chromaticsymmetricfunctionSchurpositivityclaw-freegraphlineNewtonpolytopesaturatedcounterexamplecoloring
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

A long-standing conjecture asserted that every claw-free graph has a chromatic symmetric function with nonnegative Schur coefficients. This paper exhibits two 12-vertex line graphs—automatically claw-free—whose chromatic symmetric functions have negative s_3333 coefficients, -64 and -40. It also gives a 12-vertex bipartite graph whose chromatic symmetric function is Schur-positive yet whose Newton polytope in three variables is not saturated: the color-weight vectors (6,6,0) and (8,2,2) occur, but their average (7,4,1) does not. If the computations are correct, both the 1995 claw-free conjecture and the 2018 saturation conjecture are false as stated.

What carries the argument

The chromatic symmetric function X_G(x) = sum over proper colorings κ of x_{κ(1)}...x_{κ(n)}, together with its expansion in the Schur symmetric function basis; a function is Schur-positive exactly when every Schur coefficient is nonnegative. The line graph operation L(H)—vertices are edges of H, adjacency means sharing a vertex—is the device that guarantees claw-freeness while producing graphs with nontrivial Schur behavior. For the saturation counterexample, the Newton polytope of X_G(x1,x2,x3) serves as the key object: its lattice points are the color-weight vectors arising from proper colorings, and the paper shows that two such vectors occur while their lattice average does not.

What would settle it

Independently recompute the Schur expansion of X_G for the two 12-vertex line graphs from the displayed edge lists and check whether the coefficient of s_3333 is -64 and -40; also compute the full Schur expansion of X_{G3} and exhaustively test all 3-colorings of G3 for a proper coloring with color-class sizes (7,4,1). A positive s_3333 coefficient, a nonzero m_741 coefficient, or any negative Schur coefficient for G3 would refute the corresponding claim.

Watch

Extended reading notes

Core claim

The paper centers on the chromatic symmetric function X_G, which records a monomial for every proper coloring of G, and on its expansion in the Schur basis. The authors produce two line graphs G1 and G2, obtained from 10-vertex graphs with 12 edges each, for which the s_3333 Schur coefficient of X_G is -64 and -40, respectively. Since every line graph is claw-free, these are explicit counterexamples to the claim that all claw-free graphs are Schur-positive. Independently, the authors give a 12-vertex bipartite graph G3 whose chromatic symmetric function is Schur-positive—they display the full expansion—but whose Newton polytope in three variables fails to be saturated: proper colorings with

Load-bearing premise

The counterexamples to the first conjecture rest on computer calculations of Schur expansions that are cited but not printed in the paper; if the code or the transcription of the edge lists is wrong, the negative coefficients -64 and -40 would not be established, and the Schur-positivity of G3 also depends on an unshown machine expansion.

Editorial extensions

If this is right

  • The 1995 conjecture that claw-free graphs are Schur-positive is false; claw-freeness alone does not force Schur positivity.
  • The 2018 conjecture that every Schur-positive chromatic symmetric function has saturated Newton polytope in all numbers of variables is false; the failure occurs already in three variables.
  • The 12-vertex threshold for the saturation conjecture is sharp, matching the prior lower bound from earlier work.
  • The two line-graph counterexamples lie outside the previously known Schur-positive class of claw-free incomparability graphs, so positive results for that class do not extend to all claw-free graphs.
  • The explicit edge lists give concrete test graphs for any future conjecture about claw-free Schur positivity or Newton-polytope saturation.

Reading between the lines

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

  • Editorial extension: the non-saturation of G3 at k=3 suggests that the SNP property may fail for many Schur-positive chromatic symmetric functions; a natural test is whether other small bipartite graphs with Schur-positive X_G also have unsaturated Newton polytopes.
  • Editorial extension: the minimality claim for G2 could be independently verified by exhaustive enumeration of all claw-free graphs below that size; any new small graph with a negative Schur coefficient would either confirm the reported minimality or refine it.
  • Editorial extension: the method of searching by computer for negative Schur coefficients points toward a feasible automated program: enumerate claw-free graphs by edge count, compute a finite window of Schur coefficients, and stop at the first negative one. This could reveal infinite families rather than isolated examples.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The manuscript gives two explicit 12-vertex line graphs G1 and G2, with edge lists in Examples 2.1 and 2.2, for which it claims the Schur coefficient [s_3333]X_G is -64 and -40, respectively. Since line graphs are claw-free, this would disprove Stanley's 1995 conjecture that claw-free graphs have Schur-positive chromatic symmetric functions. It also gives a 12-vertex bipartite graph G3, with edge list in Example 3.1, for which it claims X_G3 is Schur-positive but the three-variable Newton polytope is not saturated: the monomials x_6^6 x_6^6 and x_8^8 x_2^2 x_2^2 occur while their average x_7^7 x_4^4 x_1 does not, contradicting Monical's 2018 SNP conjecture for Schur-positive chromatic symmetric functions. The non-saturation argument for G3 is elementary and self-contained; the Schur positivity of G3 and the negative coefficients for G1/G2 are presented as SageMath computations.

Significance. If the computational claims are correct, the paper settles two longstanding conjectures in the negative with explicit, small, checkable graphs. The G3 non-saturation proof is elegant and can be verified by hand from the displayed edge list. The paper also situates the examples in the context of recent work by the authors and by Prajapati, and the explicit data makes independent recomputation feasible. The main weakness is that the decisive computations are only referenced as 'see code' and are not included in the manuscript, so the central disproofs are not yet reproducible from the text.

major comments (3)
  1. [Section 2, Examples 2.1 and 2.2] The claims [s_3333]X_G1=-64 and [s_3333]X_G2=-40 are the central disproof of Conjecture 1.1. The text says only 'Using SageMath (see code)' but no code, output, or certificate is provided. Since the entire counterexample rests on these two coefficients, the manuscript must include the complete SageMath script or a full Schur expansion (or another independently checkable certificate) so that a reader can verify the negative coefficients without reimplementing the computation from scratch.
  2. [Section 3, displayed Schur expansion of G3] The Schur positivity of X_G3 is load-bearing for the counterexample to Monical's conjecture, but the displayed expansion is not reproducible as written. The notation is undefined and appears corrupted: e.g. 's_112', 's_2110', 's_32215' are not standard partition labels, and the expansion is labeled 'XG2' instead of 'XG3'. The paper needs to give the actual SageMath code that produces the expansion, or a machine-readable version in a clearly defined notation, so that the positivity claim can be checked.
  3. [Section 4] The promised computational artifacts are not present. The references 'see code' and 'see example of the output here' are dangling; no code, output, or prompts are included. This is a presentation issue for Section 4, but it directly affects the verifiability of the main results because those artifacts are the only source for the G1/G2 Schur coefficients and the G3 Schur expansion.
minor comments (5)
  1. [Section 3] The label 'XG2' in the displayed Schur expansion should be 'XG3'.
  2. [Section 3, proof of no weight (7,4,1)] The argument is correct but terse. Expanding the sentence about the vertices 1,2,3,4 and the remaining outer vertex into an explicit case analysis would improve readability.
  3. [Section 3, notation] The shorthand for Schur function indices (e.g. 's_112', 's_2110') should be defined, or standard partition notation with commas/exponents should be used.
  4. [Abstract and Introduction] The phrase 'Both of these examples' is slightly misleading since the paper contains three graphs (G1 and G2 for Stanley's conjecture, G3 for Monical's). It would be clearer to say 'Both counterexamples' or 'The counterexamples'.
  5. [Section 4] The hyperlinks 'here' are placeholders. Either include the actual prompt/output files or remove them; in the current form the section promises material that is not part of the manuscript.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the counterexamples are explicit graph constructions verified by separate computations and a self-contained non-saturation argument; omitted code is a reproducibility gap, not circular reasoning.

full rationale

The paper's derivation chain contains no fitted parameter renamed as a prediction, no definition that presupposes the target conclusion, and no load-bearing chain of self-citations. The counterexamples to Stanley's conjecture are explicit graphs H1 and H2 with full edge lists; the claimed negative Schur coefficients for their line graphs are stated as SageMath computations ('Using SageMath (see code)'). Because the graphs are fully specified, these coefficients are checkable outputs, not inputs used to define the graphs, so there is no circular reduction. The fact that the code is not included is a reproducibility/verification limitation, not circularity. The counterexample to Monical's conjecture is supported by a rigorous, self-contained argument: explicit colorings establish the presence of weights (6,6,0) and (8,2,2), and a combinatorial argument shows the average weight (7,4,1) is absent; the Schur positivity of G3 is then asserted from a separately displayed SageMath expansion, not from the non-saturation proof. The citation to the authors' prior work [MMS24] appears only as contextual background about Newton polytopes and does not supply the counterexamples or any uniqueness/ansatz that forces the conclusions. No step reduces by construction to its own inputs, so the appropriate circularity score is 0.

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

The central claims depend on no fitted parameters and no newly invented entities. The only nontrivial external reliance is the correctness of exact computer algebra, which is not fully documented in the text.

assumptions (3)
  • standard math Line graphs are claw-free (West, Theorem 7.1.18).
    Used in Section 2 to conclude G1 and G2 are claw-free.
  • domain assumption The SageMath computations of Schur expansions are correct.
    Used in Section 2 for coefficients -64 and -40 and in Section 3 for the Schur-positive expansion of G3; no code or derivation is included.
  • standard math Standard definitions of support and Newton polytope saturation.
    Used in Section 3 to define the SNP property and identify the missing lattice point (7,4,1).

how reviews work

0 comments
Cite this review

Pith. "Pith review of Chromatic symmetric functions of claw-free graphs are not Schur positive." pith.science (2026). https://pith.science/paper/XU7AH4BI

@misc{pith2026260721508,
  author       = {Pith},
  title        = {Pith review of: Chromatic symmetric functions of claw-free graphs are not Schur positive},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XU7AH4BI}},
  note         = {Machine review of arXiv:2607.21508}
}
read the original abstract

Chromatic symmetric functions are well-studied symmetric functions in algebraic combinatorics that generalize chromatic polynomials of graphs. In 1995, Stanley introduced these symmetric functions and conjectured that they are Schur positive for claw-free graphs. We give examples of a line graphs, which are thus claw-free, whose chromatic symmetric function have a negative coefficient in its Schur expansion. We also give a counterexample to the 2018 conjecture of Monical that Schur positive chromatic symmetric functions have saturated Newton polytope when expanded in any finite number of variables. Both of these examples were found using ChatGPT-5.6 Sol Pro.

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. A counterexample to the claw-free Schur-positivity conjecture

    math.CO 2026-07 accept novelty 8.0 of 10

    A 12-vertex claw-free graph with chromatic symmetric function coefficient [s_(3,3,3,3)] = −64 disproves the Gasharov–Stanley Schur-positivity conjecture.

  2. Two infinite families of counterexamples to the Stanley--Gasharov conjecture

    math.CO 2026-07 conditional novelty 7.0 of 10

    Claw-free graphs, already known to disprove the Stanley--Gasharov conjecture, are shown to yield infinitely many counterexamples in both line-graph and non-line-graph families, plus minimality of the base examples.

Reference graph

Works this paper leans on

88 extracted references · 1 canonical work pages · cited by 2 Pith papers

  1. [1]

    Kodama , HOWPUBLISHED =

    T. Kodama , HOWPUBLISHED =. Is this a counterexample to the claw-free. https://mathoverflow.net/q/513515 , URL =

  2. [2]

    Prajapati , title =

    J. Prajapati , title =. GitHub repository , HOWPUBLISHED =. 2026 , publisher =

    2026

  3. [3]

    Discrete Math

    Bijective proof of a conjecture on unit interval posets , author=. Discrete Math. Theor. Comput. Sci. , FJOURNAL =. 2024 , NUMBER =

    2024

  4. [4]

    Proof of a conjecture of

    G. Proof of a conjecture of. arXiv preprint arXiv:2212.12171 , year=

    arXiv

  5. [5]

    Anari and K

    N. Anari and K. Liu and S. O. Gharan and C. Vinzant , JOURNAL =. Log-. 2024 , NUMBER =

    2024

  6. [6]

    Hikita , year=

    T. Hikita , year=. A proof of the

  7. [7]

    arXiv e-prints , keywords =

    On Macdonald expansions of q -chromatic symmetric functions and the Stanley-Stembridge Conjecture. arXiv e-prints , keywords =. doi:10.48550/arXiv.2504.06936 , archivePrefix =. 2504.06936 , primaryClass =

    arXiv doi:10.48550/arxiv.2504.06936

  8. [8]

    Birkhoff, G. D. , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1912/13 , NUMBER =. doi:10.2307/1967597 , URL =

    doi:10.2307/1967597 1912

Show all 88 references
  1. [9]

    , TITLE =

    Murota, K. , TITLE =. 2003 , PAGES =. doi:10.1137/1.9780898718508 , URL =

    2003 doi

  2. [10]

    Lorentzian polynomials , JOURNAL =

    Br\". Lorentzian polynomials , JOURNAL =. 2020 , NUMBER =. doi:10.4007/annals.2020.192.3.4 , URL =

    2020 doi

  3. [11]

    and Huh, J

    Adiprasito, K. and Huh, J. and Katz, E. , TITLE =. Ann. of Math. (2) , FJOURNAL =. 2018 , NUMBER =. doi:10.4007/annals.2018.188.2.1 , URL =

    2018 doi

  4. [12]

    and Katz, E

    Huh, J. and Katz, E. , TITLE =. Math. Ann. , FJOURNAL =. 2012 , NUMBER =. doi:10.1007/s00208-011-0777-6 , URL =

    2012 doi

  5. [13]

    , TITLE =

    Huh, J. , TITLE =. J. Amer. Math. Soc. , FJOURNAL =. 2012 , NUMBER =. doi:10.1090/S0894-0347-2012-00731-0 , URL =

    2012 doi

  6. [14]

    and Seymour, P

    Chudnovsky, M. and Seymour, P. , TITLE =. J. Combin. Theory Ser. B , FJOURNAL =. 2007 , NUMBER =. doi:10.1016/j.jctb.2006.06.001 , URL =

    2007 doi

  7. [15]

    , TITLE =

    Rado, R. , TITLE =. J. London Math. Soc. , FJOURNAL =. 1952 , PAGES =. doi:10.1112/jlms/s1-27.1.1 , URL =

    1952 doi

  8. [16]

    and Matherne, J

    Huh, J. and Matherne, J. P. and M\'. Logarithmic concavity of. Trans. Amer. Math. Soc. , FJOURNAL =. 2022 , NUMBER =. doi:10.1090/tran/8606 , URL =

    2022 doi

  9. [17]

    Stanley, R. P. , TITLE =. Adv. Math. , FJOURNAL =. 1995 , NUMBER =. doi:10.1006/aima.1995.1020 , URL =

    1995

  10. [18]

    Stanley, R. P. , TITLE =. Discrete Math. , FJOURNAL =. 1998 , NUMBER =. doi:10.1016/S0012-365X(98)00146-0 , URL =

    1998 doi

  11. [19]

    Stanley, R. P. , TITLE =. 2012 , PAGES =

    2012

  12. [20]

    Stanley, R. P. , TITLE =. 1999 , PAGES =. doi:10.1017/CBO9780511609589 , URL =

    1999 doi

  13. [21]

    and Robichaux, C

    Adve, A. and Robichaux, C. and Yong, A. , TITLE =. S\'. 2020 , PAGES =

    2020

  14. [22]

    and Robichaux, C

    Adve, A. and Robichaux, C. and Yong, A. , TITLE =. Adv. Math. , FJOURNAL =. 2021 , PAGES =

    2021

  15. [23]

    and Tokcan, N

    Monical, C. and Tokcan, N. and Yong, A. , TITLE =. Selecta Math. (N.S.) , FJOURNAL =. 2019 , NUMBER =. doi:10.1007/s00029-019-0513-8 , URL =

    2019 doi

  16. [24]

    , year =

    Monical, C. , year =. Polynomials in algebraic combinatorics , school=

  17. [25]

    , journal =

    Gasharov, V. , journal =. Incomparability graphs of (3+1)-free posets are

  18. [26]

    Stanley, R. P. and Stembridge, J. R. , journal =. On immanants of

  19. [27]

    T. Y. Chow , journal =. Descents, quasi-symmetric functions,

  20. [28]

    Linear R elations Between q - C hromatic symmetric functions

    Guay-Paquet, M. Linear R elations Between q - C hromatic symmetric functions

  21. [29]

    , journal =

    Guay-Paquet, M. , journal =. A second proof of the

  22. [30]

    and Cid-Ruiz, Y

    Castillo, F. and Cid-Ruiz, Y. and Mohammadi, F. and Monta\. Double. Forum Math. Sigma , FJOURNAL =. 2023 , PAGES =. doi:10.1017/fms.2023.101 , URL =

    2023 doi

  23. [31]

    , title = "

    Guay-Paquet, M. , title = ". arXiv e-prints , keywords =. doi:10.48550/arXiv.1306.2400 , archivePrefix =. 1306.2400 , primaryClass =

    doi arXiv

  24. [32]

    Valiant, L. G. , TITLE =. Theoret. Comput. Sci. , FJOURNAL =. 1979 , NUMBER =. doi:10.1016/0304-3975(79)90044-6 , URL =

    1979 doi

  25. [33]

    and Wachs, M

    Shareshian, J. and Wachs, M. L. , TITLE =. Adv. Math. , FJOURNAL =. 2016 , PAGES =

    2016

  26. [34]

    and Morales, A

    Guay-Paquet, M. and Morales, A. H. and Rowland, E. , TITLE =. Ann. Comb. , FJOURNAL =. 2014 , NUMBER =. doi:10.1007/s00026-014-0249-2 , URL =

    2014 doi

  27. [35]

    Stanley, R. P. , TITLE =. 2015 , PAGES =. doi:10.1017/CBO9781139871495 , URL =

    2015 doi

  28. [36]

    and Goeckner, B

    Bayer, M. and Goeckner, B. and Hong, S. J. and McAllister, T. and Olsen, M. and Pinckney, C. and Vega, J. and Yip, M. , journal=. Lattice polytopes from

  29. [37]

    Advances in Mathematics , volume=

    Schubert polynomials as integer point transforms of generalized permutahedra , author=. Advances in Mathematics , volume=. 2018 , publisher=

    2018

  30. [38]

    International Mathematics Research Notices , volume=

    Permutohedra, associahedra, and beyond , author=. International Mathematics Research Notices , volume=. 2009 , publisher=

    2009

  31. [39]

    , TITLE =

    Haglund, J. , TITLE =

  32. [40]

    Wagner, D. G. , TITLE =. Bull. Amer. Math. Soc. (N.S.) , VOLUME =. 2011 , NUMBER =

    2011

  33. [41]

    Lewis, J. B. and Zhang, Y. X , TITLE =. J. Combin. Theory Ser. A , FJOURNAL =. 2013 , NUMBER =

    2013

  34. [42]

    and Riordan, J

    Kaplansky, I. and Riordan, J. , TITLE =. Duke Math. J. , FJOURNAL =. 1946 , PAGES =

    1946

  35. [43]

    and Robins, S

    Beck, M. and Robins, S. , TITLE =. 2015 , PAGES =

    2015

  36. [44]

    and Br\"

    Borcea, J. and Br\". The. Invent. Math. , FJOURNAL =. 2009 , NUMBER =

    2009

  37. [45]

    and Br\"

    Borcea, J. and Br\". The. Comm. Pure Appl. Math. , FJOURNAL =. 2009 , NUMBER =

    2009

  38. [46]

    and Seymour, P

    Chudnovsky, M. and Seymour, P. , TITLE =. J. Combin. Theory Ser. B , VOLUME =. 2007 , NUMBER =

    2007

  39. [47]

    and Bergeron, N

    Aguiar, M. and Bergeron, N. and Sottile, F. , TITLE =. Compos. Math. , VOLUME =. 2006 , NUMBER =

    2006

  40. [48]

    and Sazdanovic, R

    Chandler, A. and Sazdanovic, R. and Stella, S. and Yip, M. , TITLE =. Adv. in Appl. Math. , FJOURNAL =. 2023 , PAGES =. doi:10.1016/j.aam.2023.102559 , URL =

    2023

  41. [49]

    A categorification of the chromatic symmetric function , author=. J. of Combin. Theory, Ser. A , volume=

  42. [50]

    McDonald, L. M. and Moffatt, I. , TITLE =. J. Stat. Phys. , FJOURNAL =. 2012 , NUMBER =

    2012

  43. [51]

    and Chow, T

    Brosnan, P. and Chow, T. Y. , TITLE =. Adv. Math. , VOLUME =. 2018 , PAGES =

    2018

  44. [52]

    and Precup, M

    Harada, M. and Precup, M. E. , TITLE =. Algebr. Comb. , VOLUME =. 2019 , NUMBER =

    2019

  45. [53]

    An efficient algorithm for deciding vanishing of Schubert polynomial coefficients , author=. Adv. in Math. , volume=

  46. [54]

    Tropicalization, symmetric polynomials, and complexity , author=. J. Symbolic Comput. , volume=

  47. [55]

    Duke Mathematical Journal , volume=

    The problem of the rooks and its applications , author=. Duke Mathematical Journal , volume=. 1946 , publisher=

    1946

  48. [56]

    Upper bounds on

    Pak, Igor and Panova, Greta , journal=. Upper bounds on

  49. [57]

    Macdonald, I. G. , TITLE =

  50. [58]

    White, D. E. , TITLE =. Adv. in Math. , VOLUME =. 1980 , NUMBER =

    1980

  51. [59]

    , TITLE =

    Gerstenhaber, M. , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1961 , PAGES =. doi:10.2307/1970297 , URL =

    1961 doi

  52. [60]

    Indagationes Mathematicae , year =

    A transitivity result for ad-nilpotent ideals in type. Indagationes Mathematicae , year =. doi:https://doi.org/10.1016/j.indag.2021.05.003 , url =

    2021 doi

  53. [61]

    and Sommers, E

    Precup, M. and Sommers, E. , title =. Pure Appl. Math. Q. , issn =. 2025 , language =. doi:10.4310/PAMQ.241203042708 , keywords =

    2025 doi

  54. [62]

    and Mellit, A

    Carlsson, E. and Mellit, A. , TITLE =. J. Amer. Math. Soc. , FJOURNAL =. 2018 , NUMBER =. doi:10.1090/jams/893 , URL =

    2018 doi

  55. [63]

    and Nigro, A

    Abreu, A. and Nigro, A. , TITLE =. J. Combin. Theory Ser. A , FJOURNAL =. 2021 , PAGES =. doi:10.1016/j.jcta.2021.105407 , URL =

    2021

  56. [64]

    Rubey and C

    M. Rubey and C. Stump and others , title =

  57. [65]

    and Sulzgruber, R

    Alexandersson, P. and Sulzgruber, R. , TITLE =. Adv. Math. , VOLUME =. 2022 , PAGES =

    2022

  58. [66]

    and Panova, G

    Alexandersson, P. and Panova, G. , TITLE =. Discrete Math. , FJOURNAL =. 2018 , NUMBER =

    2018

  59. [67]

    , TITLE =

    Schrijver, A. , TITLE =. 2003 , PAGES =

    2003

  60. [68]

    , TITLE =

    Whitney, H. , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1932 , NUMBER =

    1932

  61. [69]

    Mason, J. H. , TITLE =. Combinatorics (. 1972 , MRCLASS =

    1972

  62. [70]

    and Wilson, A

    Haglund, J. and Wilson, A. T. , TITLE =. Electron. J. Combin. , VOLUME =. 2020 , NUMBER =

    2020

  63. [71]

    , title =

    Chow, T. , title =. 2015 , note=

    2015

  64. [72]

    and Setiabrata, L

    M\'esz\'aros, K. and Setiabrata, L. , title =. Algebraic Combinatorics , pages =. 2021 , doi =

    2021

  65. [73]

    and Ono, K

    Haglund, J. and Ono, K. and Wagner, D. G. , TITLE =. Topics in

  66. [74]

    , TITLE =

    Haglund, J. , TITLE =. European J. Combin. , FJOURNAL =. 2000 , NUMBER =. doi:10.1006/eujc.2000.0422 , URL =

    2000

  67. [75]

    and Morales, A

    Colmenarejo, L. and Morales, A. H. and Panova, G. , TITLE =. European J. Combin. , VOLUME =. 2023 , PAGES =

    2023

  68. [76]

    Hardy, G. H. and Littlewood, J. E. and P\'. Inequalities , SERIES =. 1988 , PAGES =

    1988

  69. [77]

    and Scott, G

    Orellana, R. and Scott, G. , TITLE =. Discrete Math. , FJOURNAL =. 2014 , PAGES =

    2014

  70. [78]

    Foley, A. M. and Ho\`ang, C. T. and Merkel, O. D. , TITLE =. Electron. J. Combin. , FJOURNAL =. 2019 , NUMBER =. doi:10.37236/8211 , URL =

    2019 doi

  71. [79]

    and Foley, A

    Dahlberg, S. and Foley, A. and van Willigenburg, S. , title =. J. Eur. Math. Soc. (JEMS) , issn =. 2020 , doi =

    2020

  72. [80]

    Matherne, J. P. and Morales, A. H. and Selover, J. , TITLE =. Selecta Math. (N.S.) , FJOURNAL =. 2024 , NUMBER =. doi:10.1007/s00029-024-00928-4 , URL =

    2024 doi

  73. [81]

    , TITLE =

    Braun, B. , TITLE =. Recent trends in combinatorics , SERIES =

  74. [82]

    1995 , PAGES =

    The book of traces , EDITOR =. 1995 , PAGES =. doi:10.1142/9789814261456 , URL =

    1995 doi

  75. [83]

    West, D. B. , title =. 1996 , publisher =

    1996

  76. [84]

    , title =

    Haiman, M. , title =. J. Am. Math. Soc. , issn =. 1993 , doi =

    1993

  77. [85]

    and Hyatt, M

    Clearman, S. and Hyatt, M. and Shelton, B. and Skandera, M. , title =. Electron. J. Comb. , issn =. 2016 , language =

    2016

  78. [86]

    , title =

    Patrias, R. , title =. Math. Intell. , issn =. 2019 , language =. doi:10.1007/s00283-018-09862-8 , keywords =

    2019 doi

  79. [87]

    arXiv e-prints , keywords =

    A counterexample to the claw-free Schur-positivity conjecture. arXiv e-prints , keywords =

  80. [88]

    arXiv e-prints , keywords =

    An infinite family of counterexamples to the Stanley--Gasharov conjecture. arXiv e-prints , keywords =

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.