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arxiv: 2606.02257 · v1 · pith:PKRP77PBnew · submitted 2026-06-01 · 🧮 math.CO

Counterexamples to Robichaux's conjecture for Grothendieck polynomials

Avery St. Dizier This is my paper

Pith reviewed 2026-06-28 14:01 UTC · model grok-4.3

classification 🧮 math.CO MSC 05E05
keywords Grothendieck polynomialsKohnert rulespattern avoidance1432-avoiding permutationsK-theorySchubert calculuscombinatorial bijections
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0 comments X

The pith

Robichaux's ghost K-Kohnert rule for Grothendieck polynomials has counterexamples, but both it and the Ross-Yong rule hold for 1432-avoiding permutations via an explicit bijection.

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

The paper constructs permutations where Robichaux's revised ghost K-Kohnert rule fails to produce the correct Grothendieck polynomial. It then gives an explicit bijection proving that the rule, along with the original Ross-Yong conjecture, correctly computes the polynomials precisely when the permutation avoids the pattern 1432. This bijection also yields a characterization of 1432-avoidance in terms of Kohnert diagrams. A sympathetic reader cares because these rules aim to give positive combinatorial formulas in K-theoretic Schubert calculus, and identifying their exact domain of validity refines how such formulas can be applied.

Core claim

Robichaux's ghost K-Kohnert rule does not hold in general for Grothendieck polynomials, as shown by explicit counterexamples. For the class of 1432-avoiding permutations, however, an explicit bijection demonstrates that both the Ross-Yong rule and Robichaux's rule produce the correct polynomial, and this agreement supplies a Kohnert-theoretic characterization of 1432-avoidance.

What carries the argument

The explicit bijection between Ross-Yong diagrams and ghost K-Kohnert diagrams for 1432-avoiding permutations that preserves the counted weights.

If this is right

  • Both rules give the correct Grothendieck polynomial on every 1432-avoiding permutation.
  • The two rules coincide exactly on the 1432-avoiding class.
  • 1432-avoidance admits a characterization by the existence of matching Kohnert diagrams under the two rules.

Where Pith is reading between the lines

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

  • The bijection technique might extend to other pattern-avoidance classes where the rules partially agree.
  • The characterization could be used to test 1432-avoidance by checking diagram compatibility without enumerating all permutations.
  • Similar counterexample searches may be feasible for other proposed K-theoretic rules.

Load-bearing premise

The counterexamples and the bijection correctly identify violations or preservations under the precise definition of the ghost moves and the pattern-avoidance condition.

What would settle it

An independent computation of the Grothendieck polynomial for one of the claimed counterexample permutations that fails to match the value given by Robichaux's rule.

Figures

Figures reproduced from arXiv: 2606.02257 by Avery St. Dizier.

Figure 1
Figure 1. Figure 1: Note that bump and half-bump tiles are depicted in blue, first crossings in red, [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: ♢ [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: A grave diagram (left), and the results of an ordinary Kohnert move (middle) and a K-Kohnert move (right). Ross and Yong made an analogous conjecture for Lascoux polynomials, an inhomogeneous version of Demazure characters ([29, Conjecture 1.4]). Their Lascoux conjecture was proven by Pan and Yu [26] via a bijection to certain set-valued tableaux defined by Shimozono and Yu [30]. It was shown by Robichaux … view at source ↗
Figure 4
Figure 4. Figure 4: A grave diagram (left), and the results of a ghost Kohnert move (middle) and a ghost K-Kohnert move (right). We present counterexamples to Conjecture 3.7 in both numerical directions. Proposition 3.8. The permutation v = 142396857 and exponent γ = (4, 6, 3, 3, 0, 0, 0, 0, 0) witness #PD(v, γ) = 1 and #GKKoh(v, γ) ≥ 2. Proof. The Rothe diagram of v and a pipe dream of v with weight γ is shown in [PITH_FULL… view at source ↗
Figure 5
Figure 5. Figure 5: The Rothe diagram of v = 142396857 (left) and unique pipe dream of v with weight γ = (4, 6, 3, 3) (right). On the other hand, the two diagrams in [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Witnesses to #GKKoh(v, γ) ≥ 2 [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The three pipe dreams of u = 123765948 with weight β = (4, 4, 4, 4, 1). 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The Rothe diagram of u = 123765948 (left) and witnesses to #GKKoh(u, β) ≥ 2 (right) for β = (4, 4, 4, 4, 1). Proof. A direct computation of Gu gives #PD(u, β) = [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The forced construction steps in the proof of Claim 2 of Proposi￾tion 3.9. The steps (1), (2), (3), (4) occur in labeled order, with the step (1, 4) occurring any time after step (1) and before step (4). The steps (5+) must occur afterwards, but are at odds with the required end result. Since column 4 must consist of 4 boxes and 1 ghost, it is enough to show the ghost lies in row 2 or row 5. It is immediat… view at source ↗
Figure 10
Figure 10. Figure 10: The Rothe diagram of w = 123764958 (left), and a witness to #PD(w, α) = 1 (right) for α = (4, 4, 4, 3) [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: A diagram in GKKoh(w) \ KKoh(w) for w = 123764958. 4. 1432-Avoiding Permutations In [28, Theorem 5.1], Robichaux proved that both the ghost rule (Conjecture 3.7) and the Ross–Yong rule (Conjecture 3.4) hold for 321-avoiding permutations. This was done through reduction to a special case of the Pan–Yu bijection [26]. Robichaux also suggested both conjectures hold for 1432-avoiding permutations, and may adm… view at source ↗
Figure 12
Figure 12. Figure 12: The set SVRT(w) for w = 1423. The corresponding signed mono￾mials in Gw are shown below each tableau. For w 1432-avoiding, we construct a weight-preserving bijection SVRT(w) ←→ KKoh(w). 4.2. Tableaux to Grave Diagrams. Definition 4.4. For T ∈ SVRT(w) and (r, c) ∈ D(w), write T(r, c) = {a1 < a2 < · · · < am}. Define Θw(T) = (BT , GT ) by placing a box at (a1, c) and ghosts at (a2, c), . . . ,(am, c). Ex￾pl… view at source ↗
Figure 13
Figure 13. Figure 13: An example of the map Θw. Lemma 4.6. If T ∈ SVRT(w), then Θw(T) is a grave diagram [PITH_FULL_IMAGE:figures/full_fig_p012_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: An example of the map Ξw. 4.4. The Easy Inclusion. To work towards the bijection SVRT(w) KKoh(w) Θw Ξw we first prove the inclusion Θw(SVRT(w)) ⊆ KKoh(w). We begin with the following technical lemma describing the local properties of Θw within each column. Lemma 4.12. Let T ∈ SVRT(w). Choose r maximal such that some cell in row r is not filled by {r}, and choose c minimal such that T(r, c) ̸= {r}. Set p =… view at source ↗
Figure 15
Figure 15. Figure 15: A visualization of Lemma 4.15. Lemma 4.16. Fix a column c of E ∈ KKoh(w), and let column c of D(w) have boxes in rows i1 < i2 < · · · < im. Suppose that from north to south, the threads of column c of E have row sets R1, . . . , Rm. Then max(Rk) ≤ ik for all 1 ≤ k ≤ m. Proof. For E = D(w), clearly Rk = {ik} and the result is immediate. Suppose that the result holds for E, and consider a diagram E ′ ∈ KKoh… view at source ↗
Figure 16
Figure 16. Figure 16: The situation of Lemma 4.21. Lemma 4.21. Assume Setup 4.19. Suppose T satisfies all row inequalities (4.1) and T ′ does not. Then there exists an index k with 0 ≤ k < M, and a column index d > c such that (rk, d) ∈ D(w) satisfying: • max T(rk, d) = p + 1 + k; • T(rk, c) = {p + 1 + k}; and • T ′ (rk, c) = {p + k}. Proof. By Lemma 4.20, a new violation in T ′ must have the form max T(rk, d) = max T ′ (rk, d… view at source ↗
Figure 17
Figure 17. Figure 17: The bijection between K-Kohnert diagrams and set-valued Rothe tableaux of w = 1423. Example 4.27. For the (1432-avoiding) permutation w = 1423, the bijection KKoh(w) ↔ SVRT(w) is shown in [PITH_FULL_IMAGE:figures/full_fig_p022_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: The situation in the proof of Corollary 4.28. Let S = {p0} ∪ H0 ∪ {p + 1, p + 2, . . . , q} ∪ H1. Suppose (r, c) ∈ D(w) is the Rothe cell with T(r, c) = S. Then T ′ (r, c) = ( (S \ {q}) ∪ {p} for E −→ E ′ via a ghost Kohnert move, S ∪ {p} for E −→ E ′ via a ghost K-Kohnert move. It is immediate that min T ′ (r, c) = p0 = min T(r, c) and max T ′ (r, c) ≤ max T(r, c). Note T and T ′ agree on all other Rothe… view at source ↗
Figure 19
Figure 19. Figure 19: Achieving the left diagram of [PITH_FULL_IMAGE:figures/full_fig_p026_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Achieving the right diagram of [PITH_FULL_IMAGE:figures/full_fig_p027_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Achieving the left diagram of [PITH_FULL_IMAGE:figures/full_fig_p027_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Achieving the right diagram of [PITH_FULL_IMAGE:figures/full_fig_p028_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Achieving the diagram in [PITH_FULL_IMAGE:figures/full_fig_p028_23.png] view at source ↗
read the original abstract

Ross and Yong conjectured a $K$-theoretic Kohnert rule for Grothendieck polynomials. Robichaux exhibited a counterexample to the Ross--Yong rule and proposed a revised ghost $K$-Kohnert rule, proving both rules hold for 321-avoiding permutations. We provide counterexamples to Robichaux's rule and give an explicit bijection showing that both the Ross--Yong and Robichaux rules hold for 1432-avoiding permutations. As an application, we provide a Kohnert-theoretic characterization of 1432-avoidance.

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

0 major / 2 minor

Summary. The paper supplies explicit counterexamples to Robichaux's ghost K-Kohnert rule for Grothendieck polynomials. It constructs an explicit bijection proving that both the Ross-Yong and Robichaux rules hold on the class of 1432-avoiding permutations, and derives from this a Kohnert-theoretic characterization of 1432-avoidance.

Significance. The explicit counterexamples and bijection provide direct, falsifiable combinatorial evidence that advances the program of finding K-theoretic Kohnert rules. By isolating a pattern class on which the two rules coincide and characterizing it combinatorially, the work supplies concrete data that can be used to test or refine future conjectures in the K-theory of flag varieties.

minor comments (2)
  1. §2: the statement of the ghost K-Kohnert rule would benefit from an additional small example that explicitly marks the 'ghost' cells so readers can verify the rule application without consulting the original Robichaux reference.
  2. Figure 4: the permutation diagram and the corresponding Kohnert diagram are not labeled with the same indexing convention used in the surrounding text; adding consistent labels would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment and recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central results consist of explicit counterexamples to Robichaux's ghost K-Kohnert rule and an explicit bijection establishing that both the Ross-Yong and Robichaux rules hold on 1432-avoiding permutations, plus a resulting characterization of that class. These are direct, falsifiable combinatorial constructions whose validity rests only on faithful application of the stated definitions; no load-bearing step reduces by construction to a fitted parameter, self-citation chain, or self-definitional loop. The argument is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard combinatorial definitions of pattern avoidance and Kohnert diagrams; no free parameters, invented entities, or ad-hoc axioms are indicated in the abstract.

axioms (2)
  • standard math Pattern avoidance for permutations is defined via the standard notion of classical pattern containment in S_n.
    Used to define the 321-avoiding and 1432-avoiding classes mentioned in the abstract.
  • domain assumption Kohnert diagrams and their K-theoretic variants are well-defined combinatorial objects with established generating functions for Grothendieck polynomials.
    Invoked implicitly when stating that the rules 'hold' for certain classes.

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