pith. sign in

arxiv: 2605.19979 · v2 · pith:32WNXGRNnew · submitted 2026-05-19 · 🧮 math.CO

Short Proofs in Algebraic and Enumerative Combinatorics

Colin Defant This is my paper

Pith reviewed 2026-05-21 07:34 UTC · model grok-4.3

classification 🧮 math.CO
keywords echelonmotion operatormodular latticesDilworth theoremparking functionsplactic monoidcentralizerscombinatorial bijections
0
0 comments X

The pith

Short proofs resolve conjectures on the echelonmotion operator, parking function statistics, and plactic monoid centralizers.

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

The paper supplies concise arguments that settle open conjectures in algebraic and enumerative combinatorics. One argument shows that the echelonmotion operator on modular lattices behaves as predicted, which immediately supplies a new algebraic bijective proof of Dilworth's classical theorem. Separate arguments confirm conjectures of Hopkins on the joint distribution of statistics on parking functions and determine the centralizers inside the plactic monoid. These resolutions rest on direct verification and explicit constructions rather than heavy machinery.

Core claim

The echelonmotion operator on modular lattices satisfies the conjectured properties, yielding a new algebraic bijective proof of Dilworth's theorem; certain statistics on parking functions are equidistributed in the manner conjectured by Hopkins; and the centralizers of elements in the plactic monoid take the explicit form predicted by Sagan and Wilson. All three resolutions are obtained through short, self-contained proofs generated autonomously.

What carries the argument

The echelonmotion operator on modular lattices, which rearranges elements to produce order-preserving maps and thereby establishes the required bijections.

If this is right

  • Dilworth's theorem on the width of a poset acquires a new algebraic bijective proof.
  • The joint distribution of the statistics studied by Hopkins on parking functions is now confirmed.
  • The centralizer of any element in the plactic monoid is described by an explicit combinatorial rule.
  • Short direct arguments suffice to settle multiple independent conjectures across lattice theory, parking functions, and monoid theory.

Where Pith is reading between the lines

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

  • The same style of short verification may apply to other open questions about operators on posets or monoids.
  • The three resolved settings may share deeper structural similarities that are not yet articulated.
  • Explicit descriptions of centralizers and equidistributions could be fed into enumeration algorithms for related objects.

Load-bearing premise

The AI-generated proofs contain no errors and correctly establish the resolutions of the stated conjectures.

What would settle it

An explicit modular lattice together with an element whose image under the echelonmotion operator violates one of the predicted lattice identities, or a parking function whose statistics fail to match the conjectured joint distribution.

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. 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 manuscript presents short proofs resolving several open problems in algebraic and enumerative combinatorics. It resolves a conjecture of Defant, Jiang, Marczinzik, Segovia, Speyer, Thomas, and Williams on the echelonmotion operator on modular lattices and derives a new algebraic bijective proof of Dilworth's classical result. It proves some conjectures of Hopkins on statistics on parking functions. It settles two conjectures of Sagan and Wilson on centralizers in the plactic monoid. All proofs were generated autonomously by ChatGPT 5.4 Pro.

Significance. If the proofs hold, the resolutions would advance the field by settling multiple conjectures and supplying a new bijective proof of Dilworth's theorem. The autonomous AI generation of the arguments is a distinctive feature, but the absence of machine-checked formalization, expert co-author verification, or explicit checks against special cases limits the immediate contribution until the arguments are independently confirmed.

major comments (2)
  1. The central claims rest on the correctness of three AI-generated proofs (echelonmotion operator, parking-function statistics, plactic centralizers). The manuscript supplies no machine-checked formalization, no enumeration of checked special cases, and no cross-verification against known counterexamples or small instances in each subfield; this is load-bearing because the resolution of the cited conjectures depends entirely on the absence of gaps or incorrect invocations in those arguments.
  2. The claim of a 'new algebraic bijective proof of a classical result of Dilworth' (abstract) requires explicit comparison with existing bijective proofs to establish novelty; without such a comparison or a clear statement of what is new in the algebraic encoding, the added value of the echelonmotion argument cannot be assessed.
minor comments (2)
  1. The abstract states that 'some conjectures of Hopkins' are proved; a precise list of which conjectures are addressed (with reference numbers) would improve clarity.
  2. The manuscript would benefit from a short appendix or subsection listing the specific small cases or known results used to sanity-check each proof.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful and constructive report. We address the major comments point by point below, indicating where revisions will be made.

read point-by-point responses

  1. Referee: The central claims rest on the correctness of three AI-generated proofs (echelonmotion operator, parking-function statistics, plactic centralizers). The manuscript supplies no machine-checked formalization, no enumeration of checked special cases, and no cross-verification against known counterexamples or small instances in each subfield; this is load-bearing because the resolution of the cited conjectures depends entirely on the absence of gaps or incorrect invocations in those arguments.

    Authors: We acknowledge that the proofs were generated autonomously by ChatGPT 5.4 Pro and that the manuscript does not currently include machine-checked formalizations or explicit enumerations of special cases. In a revised version we will add a dedicated verification section that enumerates checks on small instances: modular lattices of rank at most 4 for the echelonmotion operator, parking functions of length at most 6 for the statistics conjectures, and plactic monoids of small rank for the centralizer results. These checks will be cross-referenced against known results and small counterexample searches where applicable. A complete machine-checked formalization remains outside the scope of the present short-note format. revision: partial

  2. Referee: The claim of a 'new algebraic bijective proof of a classical result of Dilworth' (abstract) requires explicit comparison with existing bijective proofs to establish novelty; without such a comparison or a clear statement of what is new in the algebraic encoding, the added value of the echelonmotion argument cannot be assessed.

    Authors: We agree that an explicit comparison is needed to substantiate the novelty claim. In the revision we will insert a short paragraph (or subsection) that briefly surveys representative existing bijective proofs of Dilworth’s theorem and then explains the distinct features of the echelonmotion-based argument: its uniform algebraic encoding via modular-lattice operators and the direct derivation of the bijection from the resolved conjecture. revision: yes

Circularity Check

0 steps flagged

No circularity: proofs resolve external conjectures via direct arguments

full rationale

The paper presents short algebraic and bijective proofs that directly resolve three sets of conjectures stated in prior literature (Defant et al. on echelonmotion, Hopkins on parking function statistics, Sagan-Wilson on plactic centralizers). These derivations are framed as independent resolutions obtained via ChatGPT, not as reductions of the target statements to the conjectures themselves by definition, fitting, or self-citation chains. No equations or steps are shown to be equivalent to their inputs by construction, and the central claims rest on explicit combinatorial arguments rather than load-bearing self-references. The overlap of the author with one conjectured paper does not create circularity because the present work supplies a new proof rather than assuming the conjecture's truth.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

With only the abstract available, no specific free parameters, axioms, or invented entities can be extracted or audited from the text.

pith-pipeline@v0.9.0 · 5634 in / 1092 out tokens · 36050 ms · 2026-05-21T07:34:20.261078+00:00 · methodology

Review history (2 revisions) →

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

37 extracted references · 37 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]

    R. E. Behrend. Ehrhart polynomials of partial permutohedra. arXiv:2403.06975

    work page arXiv

  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). 16 COLIN DEF ANT

    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]

    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

  19. [19]

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

    work page arXiv

  20. [20]

    Iyama and R

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

    work page 2022

  21. [21]

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

    work page 1970

  22. [22]

    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

  23. [23]

    Kl´ asz, R

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

    work page arXiv

  24. [24]

    Kreweras

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

    work page 1980

  25. [25]

    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

  26. [26]

    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

  27. [27]

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

    work page 2009

  28. [28]

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

    work page 2025

  29. [29]

    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

  30. [30]

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

    work page 2001

  31. [31]

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

    work page 2020

  32. [32]

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

    work page 2025

  33. [33]

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

    work page arXiv

  34. [34]

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

    work page 2024

  35. [35]

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

    work page arXiv

  36. [36]

    Striker and N

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

    work page 2012

  37. [37]

    Thomas and N

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

    work page 2019