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arxiv: 2606.11335 · v1 · pith:KGJPPK2Mnew · submitted 2026-06-09 · 🧮 math.CO

Extremal Matchings and Height Functions

Nickolas Anderson , Gregg Musiker This is my paper

Pith reviewed 2026-06-27 12:17 UTC · model grok-4.3

classification 🧮 math.CO
keywords almost perfect matchingsheight functionsplabic graphsdistributive latticeextremal matchingsboundary conditionspositroid cellsplanar bipartite graphs
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The pith

Almost perfect matchings on plabic graphs admit an explicit height-function construction of their extremal elements for any boundary condition.

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

The paper extends the distributive lattice on almost perfect matchings with fixed boundary condition by supplying an explicit construction of the extremal matchings expressed through height functions on the faces of the graph. It proves that the same construction produces every possible boundary condition, not only the special cases linked to positroid face labels. A reader would care because these matchings enter the boundary measurement map that parameterizes positroid cells, so a uniform height-function description could give a single way to generate the lattice across all boundaries.

Core claim

The authors give an explicit construction of extremal matchings in terms of height functions and demonstrate that this construction yields all possible boundary conditions of an almost perfect matching.

What carries the argument

Height functions that assign integer values to the faces of the embedded plabic graph and thereby select the edges of an extremal almost perfect matching.

If this is right

  • Every boundary condition on an almost perfect matching arises from some height-function labeling.
  • The extremal elements of the lattice are realized uniformly by the same height-function rule.
  • The construction does not require the boundary condition to coincide with a positroid face label.

Where Pith is reading between the lines

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

  • The uniform construction might let one generate the full lattice without first fixing a boundary condition.
  • Similar height-function rules could be tested on other planar bipartite graphs that carry distributive lattices on their matchings.

Load-bearing premise

The distributive lattice structure already known for almost perfect matchings with fixed boundary condition stays compatible with the new height-function labeling no matter which boundary condition is chosen.

What would settle it

A concrete counterexample would be any plabic graph together with a boundary condition for which the height-function construction fails to recover the top or bottom element of the lattice.

Figures

Figures reproduced from arXiv: 2606.11335 by Gregg Musiker, Nickolas Anderson.

Figure 1
Figure 1. Figure 1: (Left): Parallel edge reduction (R1); (Middle): Leaf reduction (R2); (Right): Dipole reduction [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (Left): Square move (M1); (Right): Bivalent vertex contraction and expansion (M2). [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (Left): A white lollipop; (Right): A black lollipop. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: also gives an example of strands overlaid on G0. We consider each strand up to homotopy and obtain the following consequences from the reduced property of G. Each strand τ has a unique origin and unique terminal boundary vertex with no self-intersections [Pos06, Theorem 13.2, Corollary 14.2]. For any strand τ we write ⃗τi (or simply ⃗i) to emphasize that τ terminates at the ith boundary vertex and τ i⃗ (or… view at source ↗
Figure 5
Figure 5. Figure 5: (Left): The resonance property at a white vertex [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: (Left): The perfect orientation with sink set [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: All matchings with boundary condition {2, 5, 6} on GB. From top left to bottom right, across rows, label these matchings as M1, M2, M3, M4, and M5. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: (Left): Swiveling (with direction) at a square face; (Right): Swiveling (with direction) at a [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The poset (M256, ≤) on the matchings given in [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The set of all fundamental height functions on source-labeled [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The set of all fundamental height functions on target-labeled [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: (Left): Tmax {φ1, φ4(1)} = tmax {φ1, φ4, φ5} on source-labeled GB; (Right): The land bridge L(φ1, φ4) (in red) with the heights and matched edges corresponding to φ1 and φ4 in violet and teal, at the top and bottom of GB, respectively. and φjm, we say their relative translation is the sum rℓ,m = X ℓ≤i<m ai = tm−1 −tℓ−1. Note that in bound￾ing the largest translation factor tp−1 < k, we ensure (φjp − tp−1n… view at source ↗
Figure 13
Figure 13. Figure 13: (Left): Tmin {φ2, φ6(2)} = tmin {φ2, φ5, φ6} on target-labeled GB; (Right): The land bridge L(φ2, φ6) (in red) with the heights and matched edges corresponding to φ2 and φ6 in violet and teal, at the bottom right and top left of GB, respectively. oriented path along all H-strands such taht at any point along this path there are exactly (a − 1) of the H-strands lying to its right. Now for any f ∈ L(φjℓ , φ… view at source ↗
Figure 14
Figure 14. Figure 14: The matching Tmax {φ0, φ3(1), φ7(1)} = tmax {φ0, φ3, φ7} on a source-labeled plabic graph corresponding to the top-cell of Gr(3, 9). The fundamental height regions F0, F3, and F7 are designated by violet, blue, and teal, in the top center, center right, and bottom left, respectively. In particular, the red edge in this matching does not belong to any Fj . Remark 5.22. When applying Theorem 5.20 and Theore… view at source ↗
Figure 15
Figure 15. Figure 15: (Left): M− from Tmax {φ1, φ4(1)} = tmax {φ1, φ4, φ5} on source-labeled GB and the remaining poset; (Right):M+ from Tmin {φ2, φ6(2)} = tmin {φ2, φ5, φ6} on target-labeled GB and the remaining poset. The labels of swiveled faces are included as labels on edges indicating cover relations. Example 5.24. We obtain the lattice M256 on GB of [PITH_FULL_IMAGE:figures/full_fig_p024_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: The corresponding quivers (in red) with respect to source- (left) and target-labeled (right) GB. A.2 Extremal matchings in twist maps and F-polynomials The twist map (or twist automorphism), τ , was originally introduced in [BFZ96] in the study of total positivity of unipotent cells in simple algebraic groups of simply-laced Lie type. It turns out τ is a cluster algebra automorphism. When applied to Pl¨uc… view at source ↗
Figure 17
Figure 17. Figure 17: The lattice of M256 on target-labeled GB with the yM monomials next to each matching M in this lattice. References [And25] Nickolas Anderson. Minimal matchings from height functions. Master’s thesis, University of Minnesota, 2025. [BFZ96] Arkady Berenstein, Sergey Fomin, and Andrei Zelevinsky. Parametrizations of canonical bases and totally positive matrices. Adv. Math., 122(1):49–149, 1996. [CGG+25] Roge… view at source ↗
read the original abstract

This paper studies a lattice structure for almost perfect matchings on certain planar, bipartite (plabic) graphs embedded in a disk. Postnikov's boundary measurement map, and subsequent related work, yielded that plabic graphs parameterize positroid cells within the totally nonnegative Grassmannian with the map itself given in terms of almost perfect matchings with fixed boundary condition. For finite planar bipartite graphs, Propp introduced a distributive lattice structure on their set of perfect matchings. Subsequently Muller--Speyer, provided this distributive lattice structure on the aforementioned almost perfect matchings with fixed boundary condition. Their work also identified the extremal matchings of this lattice for boundary conditions that coincide with face labels of the plabic graph given by the positroid structure. We extend this by giving an explicit construction of extremal matchings in terms of height functions and show that all possible boundary conditions of an almost perfect matching can be obtained within this construction.

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 extends the distributive lattice on almost perfect matchings of plabic graphs with fixed boundary conditions (Muller-Speyer) by supplying an explicit height-function construction of the extremal elements and proving that every possible boundary condition arises inside this construction.

Significance. If correct, the explicit height-function realization supplies a concrete, computable description of extremal matchings for arbitrary boundaries, strengthening the link between the lattice structure and the boundary-measurement parametrization of positroid cells.

minor comments (2)
  1. [Abstract] Abstract states the existence of an explicit construction but supplies neither the definition of the height function nor a proof outline; a one-sentence description of the labeling rule would make the central claim immediately verifiable from the abstract.
  2. The compatibility of the new height-function labeling with the Muller-Speyer partial order for non-fixed boundaries is asserted but not isolated as a separate lemma; a short dedicated paragraph or proposition would clarify the logical structure.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending minor revision. The report contains no specific major comments to address.

Circularity Check

0 steps flagged

No significant circularity; construction extends external lattice result

full rationale

The paper's core contribution is an explicit height-function construction for extremal almost perfect matchings that recovers arbitrary boundary conditions. It invokes the distributive lattice structure only from the cited Muller-Speyer work on the fixed-boundary case, treating that lattice as an independent external input rather than deriving or re-proving it internally. No equations reduce a claimed prediction to a fitted parameter by construction, no uniqueness theorem is imported from the authors' own prior work, and no ansatz is smuggled via self-citation. The derivation chain therefore remains self-contained against the external benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no free parameters, axioms, or invented entities are visible. The work relies on the existence of the distributive lattice from prior literature and on standard properties of height functions on planar graphs.

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Reference graph

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