pith. sign in

arxiv: 2605.19979 · v1 · pith:32WNXGRNnew · submitted 2026-05-19 · 🧮 math.CO

Short Proofs in Algebraic and Enumerative Combinatorics

Colin Defant This is my paper

Pith reviewed 2026-05-20 03:35 UTC · model grok-4.3

classification 🧮 math.CO
keywords echelonmotion operatormodular latticesDilworth theoremparking functionsplactic monoidpartial permutohedraEhrhart polynomialsbijective proofs
0
0 comments X p. Extension
pith:32WNXGRN Add to your LaTeX paper What is a Pith Number? 
\usepackage{pith}
\pithnumber{32WNXGRN}

Prints a linked pith:32WNXGRN badge after your title and writes the identifier into PDF metadata. Compiles on arXiv with no extra files. Learn more

The pith

Short proofs resolve a conjecture on the echelonmotion operator in modular lattices and supply a new algebraic bijective proof of Dilworth's theorem.

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

The paper supplies concise proofs for several open conjectures in algebraic and enumerative combinatorics. It resolves a conjecture about the echelonmotion operator on modular lattices, which in turn yields an algebraic bijective proof of Dilworth's classical result that the size of the largest antichain equals the minimum number of chains needed to cover the poset. The work also proves conjectures on statistics for parking functions, settles two conjectures on centralizers in the plactic monoid, and confirms an exact formula for the Ehrhart polynomials of partial permutohedra. A sympathetic reader would care because these results connect classical poset theorems to newer combinatorial objects through direct algebraic arguments.

Core claim

The paper establishes short proofs that resolve the conjecture on the echelonmotion operator on modular lattices, thereby obtaining a new algebraic bijective proof of Dilworth's result, while also proving conjectures on parking functions, settling two conjectures on centralizers in the plactic monoid, and verifying an exact formula for Ehrhart polynomials of partial permutohedra.

What carries the argument

The echelonmotion operator on modular lattices, which acts to establish the resolved conjecture and to construct the algebraic bijective proof of Dilworth's theorem.

If this is right

  • The echelonmotion operator on modular lattices satisfies the properties required by the resolved conjecture.
  • Dilworth's theorem on posets admits an algebraic bijective proof in addition to its classical formulations.
  • The conjectured statistics on parking functions hold as stated.
  • The two conjectures on centralizers in the plactic monoid are true.
  • The Ehrhart polynomials of partial permutohedra equal the conjectured exact formula.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The short-proof approach may extend to other open questions involving operators on lattices or monoids.
  • The bijective proof of Dilworth's theorem could be adapted to weighted posets or to related decomposition theorems.
  • Similar algebraic methods might connect parking-function statistics to other lattice operators beyond those considered here.

Load-bearing premise

That the short proofs contain no errors, gaps, or invalid steps and correctly establish the resolutions of the conjectures.

What would settle it

A small modular lattice in which the echelonmotion operator fails to produce the expected chain decomposition or antichain size matching the conjecture, or a partial permutohedron whose Ehrhart polynomial differs from the stated closed formula at a specific degree.

Figures

Figures reproduced from arXiv: 2605.19979 by Colin Defant.

Figure 1
Figure 1. Figure 1: A modular lattice labeled by two different linear extensions. The pink arrows represent echelonmotion with respect to the given linear extension. When R is a distributive lattice, Kl´asz, Marczinzik, and Thomas proved that Echσ coincides with the classical rowmotion operator on R [24]. In particular, Echσ is independent of σ when R is distributive. This motivated Defant, Jiang, Marczinzik, Segovia, Speyer,… view at source ↗
Figure 2
Figure 2. Figure 2: The rook placement from Example 3.5. Example 3.5. Let n = 6 and k = 3. Fix b = (1, 1, 2, 4, 5, 6) ∈ PF≤(6). The 3-rook placement R = {(1, 3),(2, 6),(4, 5)}, with the board Bb shaded in cyan, appears in [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
read the original abstract

We present several short proofs that resolve open problems from the algebraic and enumerative combinatorics literature. First, we consider the echelonmotion operator on modular lattices. We resolve a conjecture of Defant, Jiang, Marczinzik, Segovia, Speyer, Thomas, and Williams and, consequently, obtain a new algebraic bijective proof of a classical result of Dilworth. Second, we consider statistics on parking functions studied by Stanley and Yin and by Hopkins. We prove some conjectures of Hopkins. Third, we consider centralizers in the plactic monoid. We settle two conjectures of Sagan and Wilson. Fourth, we consider Ehrhart polynomials of partial permutohedra, which were introduced by Heuer and Striker. We verify an exact formula conjectured by Behrend, Castillo, Chavez, Diaz-Lopez, Escobar, Harris, and Insko. All of these proofs were obtained autonomously by ChatGPT 5.4 Pro.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper presents short proofs resolving several open conjectures in algebraic and enumerative combinatorics, all generated autonomously by ChatGPT 5.4 Pro. It resolves a conjecture of Defant, Jiang, Marczinzik, Segovia, Speyer, Thomas, and Williams on the echelonmotion operator on modular lattices, yielding a new algebraic bijective proof of Dilworth's theorem; proves conjectures of Hopkins on parking-function statistics; settles two conjectures of Sagan and Wilson on centralizers in the plactic monoid; and verifies an exact Ehrhart polynomial formula for partial permutohedra conjectured by Behrend et al.

Significance. If the derivations are correct, the work would be significant for delivering concise resolutions to multiple distinct conjectures, including a new bijective proof of Dilworth's classical result. The algebraic approach to the echelonmotion operator and the explicit verifications of the other statements would strengthen connections across lattice theory, parking functions, monoid theory, and Ehrhart theory. However, the absence of any reported independent verification, formalization, or spot-checks against small cases substantially weakens the immediate contribution.

major comments (2)
  1. Abstract: The resolutions of the four listed conjectures are asserted without accompanying verification details, error checks, or small-case computations. Since the proofs are the sole support for the central claims, this omission is load-bearing and prevents assessment of whether the algebraic arguments contain gaps or invalid steps.
  2. The section presenting the echelonmotion operator and Dilworth proof: the claimed algebraic bijective argument must be checked for correct application of lattice properties and bijective correspondences; no such verification or cross-reference to prior work on modular lattices is supplied.
minor comments (2)
  1. The manuscript should explicitly restate each conjecture (with citation) immediately before the corresponding proof for clarity.
  2. Add a brief note on the generation process and any post-generation human review performed, if any.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. We address each major comment below and have revised the manuscript to incorporate additional verification details and cross-references.

read point-by-point responses

  1. Referee: Abstract: The resolutions of the four listed conjectures are asserted without accompanying verification details, error checks, or small-case computations. Since the proofs are the sole support for the central claims, this omission is load-bearing and prevents assessment of whether the algebraic arguments contain gaps or invalid steps.

    Authors: We agree that the original presentation would benefit from explicit verification aids. In the revised manuscript we have added a dedicated verification subsection that includes small-case computations for each of the four results, together with direct comparisons to known tabulated values from the literature. These checks are now referenced from the abstract. revision: yes

  2. Referee: The section presenting the echelonmotion operator and Dilworth proof: the claimed algebraic bijective argument must be checked for correct application of lattice properties and bijective correspondences; no such verification or cross-reference to prior work on modular lattices is supplied.

    Authors: We have expanded the echelonmotion section with explicit citations to Defant et al. and to standard references on modular lattices. We now include a step-by-step verification of the bijective correspondences, showing how each lattice property is applied and confirming that the resulting map is indeed a bijection that recovers Dilworth’s theorem. revision: yes

Circularity Check

0 steps flagged

Minor self-citation in conjecture statement but central proofs remain independent algebraic arguments

full rationale

The manuscript resolves multiple external conjectures via short proofs generated autonomously by ChatGPT, including one conjecture co-authored by the present author. This constitutes a minor self-citation that is not load-bearing, as the paper supplies new algebraic bijective arguments rather than deriving results from the conjecture itself or from fitted parameters. No self-definitional reductions, ansatz smuggling, or renaming of known results appear in the derivation chain; the claims rest on the correctness of the presented proofs, which are treated as independent of the input conjectures.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard definitions and theorems from order theory, monoid theory, and Ehrhart theory already present in the cited literature; no new free parameters, ad-hoc axioms, or invented entities are indicated in the abstract.

axioms (1)
  • standard math Standard properties of modular lattices and the statement of Dilworth's theorem
    Invoked when deriving the new bijective proof from the echelonmotion operator result.

pith-pipeline@v0.9.0 · 5684 in / 1379 out tokens · 60355 ms · 2026-05-20T03:35:08.618737+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

39 extracted references · 39 canonical work pages · 1 internal anchor

  1. [1]

    Alexeev, M

    B. Alexeev, M. Putterman, M. Sawhney, M. Sellke, and G. Valiant. Short proofs in combinatorics and number theory. arXiv:2603.29961

    work page arXiv

  2. [2]

    Short proofs in combinatorics, probability and number theory II

    B. Alexeev, M. Putterman, M. Sawhney, M. Sellke, and G. Valiant. Short proofs in combinatorics, probability and number theory II. arXiv:2604.06609

    work page internal anchor Pith review Pith/arXiv arXiv

  3. [3]

    N. Alon. Problems and results in extremal combinatorics—I. Discrete Math., 273 (2003), 31–53

    work page 2003

  4. [4]

    N. Alon. Problems and results in extremal combinatorics—II. Discrete Math., 308 (2008), 4460–4472

    work page 2008

  5. [5]

    N. Alon. Problems and results in extremal combinatorics—III. J. Comb., 7 (2016), 319–337

    work page 2016

  6. [6]

    Armstrong, C

    D. Armstrong, C. Stump, and H. Thomas. A uniform bijection between nonnesting and noncrossing partitions. Trans. Amer. Math. Soc., 365 (2013), 4121–4151

    work page 2013

  7. [7]

    E. Barnard. The canonical join complex. Electron. J. Combin., 26 (2019)

    work page 2019

  8. [8]

    Beck and S

    M. Beck and S. Robins. Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra, second edition, Undergraduate Texts in Mathematics, Springer, 2015

    work page 2015

  9. [9]

    R. E. Behrend, F. Castillo, A. Chavez, A. Diaz-Lopez, L. Escobar, P. E. Harris, and E. Insko. Partial permu- tohedra. Discrete Comput. Geom., (2025)

    work page 2025

  10. [10]

    Chow and W

    C. Chow and W. C. Shiu. Counting simsun permutations by descents. Ann. Combin., 15 (2011), 625–635

    work page 2011

  11. [11]

    Conlon, J

    D. Conlon, J. Fox, and B. Sudakov. Short proofs of some extremal results. Combin. Prob. Comput., 23 (2014), 8–28

    work page 2014

  12. [12]

    Conlon, J

    D. Conlon, J. Fox, and B. Sudakov. Short proofs of some extremal results II. J. Combin. Theory Ser. B , 116 (2016), 173–196

    work page 2016

  13. [13]

    Defant, Y

    C. Defant, Y. Jiang, R. Marczinzik, A. Segovia, D. E Speyer, H. Thomas, and N. Williams. Rowmotion and echelonmotion. arXiv:2507.18230

    work page arXiv

  14. [14]

    Defant and N

    C. Defant and N. Williams. Semidistrim lattices. Forum Math. Sigma, 11 (2023)

    work page 2023

  15. [15]

    M. M. Deza and K. Fukuda. Loops of clutters. In Coding Theory and Design Theory: Part I Coding Theory , 72–92. Springer, 1990

    work page 1990

  16. [16]

    R. P. Dilworth. Proof of a conjecture on finite modular lattices. Ann. of Math. , 60 (1954), 359–364

    work page 1954

  17. [17]

    C. Greene. An extension of Schensted’s theorem. Adv. Math., 14 (1974), 254–265

    work page 1974

  18. [18]

    Heuer and J

    D. Heuer and J. Striker. Partial permutation and alternating sign matrix polytopes. SIAM J. Discrete Math. , 36 (2022), 2863–2888

    work page 2022

  19. [19]

    S. Hopkins. Order polynomial product formulas and poset dynamics. In Open problems in algebraic combina- torics, volume 110 of Proc. Sympos. Pure Math. , 135–157. Amer. Math. Soc., 2024

    work page 2024

  20. [20]

    S. Hopkins. Two t-analogues of the tree inversion enumerator. arXiv:2510.22385

    work page arXiv

  21. [21]

    Iyama and R

    O. Iyama and R. Marczinzik. Distributive lattices and Auslander regular algebras. Adv. Math., 398 (2022)

    work page 2022

  22. [22]

    D. E. Knuth. Permutations, matrices, and generalized Young tableaux. Pacific J. Math. , 34 (1970), 709–727

    work page 1970

  23. [23]

    Kl´ asz, M

    V. Kl´ asz, M. Kleinau, and R. Marczinzik. Classification of Auslander–Gorenstein monomial algebras: The acyclic case. arXiv:2604.02146

    work page arXiv

  24. [24]

    Kl´ asz, R

    V. Kl´ asz, R. Marczinzik, and H. Thomas. Auslander regular algebras and Coxeter matrices. arXiv:2501.09447

    work page arXiv

  25. [25]

    Kreweras

    G. Kreweras. Une famille de polynˆ omes ayant plusieurs propri´ et´ es ´ enumeratives.Period. Math. Hungar. , 11 (1980), 309–320

    work page 1980

  26. [26]

    La Ricerca Scientifica

    A. Lascoux and M.-P. Sch¨ utzenberger. Le mono¨ ıde plaxique. In Noncommutative structures in algebra and geometric combinatorics, Quaderni de “La Ricerca Scientifica” 109, CNR, 1981, 129–156

    work page 1981

  27. [27]

    Marczinzik, H

    R. Marczinzik, H. Thomas, and E. Yıldırım. On the interaction of the Coxeter transformation and the row- motion bijection. J. Comb. Algebra, 8 (2024), 359–374

    work page 2024

  28. [28]

    D. I. Panyushev. On orbits of antichains of positive roots. European J. Combin., 30 (2009), 586–594

    work page 2009

  29. [29]

    T. K. Petersen and Y. Zhuang. Zig-zag Eulerian polynomials. European J. Combin., 124 (2025)

    work page 2025

  30. [30]

    Postnikov

    A. Postnikov. Permutohedra, associahedra, and beyond. Int. Math. Res. Not. IMRN , 2009 (2009), 1026–1106

    work page 2009

  31. [31]

    T. Roby. Dynamical algebraic combinatorics and the homomesy phenomenon. In Recent trends in combina- torics, volume 159 of IMA Vol. Math. Appl. , 619–652. Springer, 2016

    work page 2016

  32. [32]

    B. E. Sagan. The Symmetric Group, second edition, Graduate Texts in Mathematics 203, Springer, 2001

    work page 2001

  33. [33]

    B. E. Sagan. Combinatorics: The Art of Counting, Graduate Studies in Mathematics 210, American Mathe- matical Society, 2020

    work page 2020

  34. [34]

    B. E. Sagan and A. N. Wilson. Centralizers in the plactic monoid. Semigroup Forum, 110 (2025), 724–744

    work page 2025

  35. [35]

    B. E. Sagan and C. Zhao. Properties of plactic monoid centralizers. arXiv:2512.21401

    work page arXiv

  36. [36]

    R. P. Stanley. Enumerative Combinatorics, Volume 2, second edition, Cambridge Studies in Advanced Math- ematics 208, Cambridge University Press, 2024

    work page 2024

  37. [37]

    R. P. Stanley and M. Yin. Some enumerative properties of parking functions. arXiv:2306.08681

    work page arXiv

  38. [38]

    Striker and N

    J. Striker and N. Williams. Promotion and rowmotion. European J. Combin., 33 (2012), 1919–1942

    work page 2012

  39. [39]

    Thomas and N

    H. Thomas and N. Williams. Rowmotion in slow motion. Proc. Lond. Math. Soc., 119 (2019), 1149–1178

    work page 2019