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A canonical realization of the alt $\nu$-associahedron

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Pith's one-line read The alt ν-Tamari lattice has its Hasse diagram as the 1-skeleton of a polytopal complex induced by a tropical hyperplane arrangement.

desk verdict The paper gives a tropical hyperplane realization of the alt ν-associahedron via areas below paths, generalizing the prior ν case and recovering Loday, but the edge match to the Hasse diagram appears to inherit the poset from earlier combinatorial work. read the letter →

arxiv 2401.17204 v2 pith:UUOENENQ submitted 2024-01-30 math.CO

Cesar Ceballos This is my paper · ORCID
classification math.CO
keywords altν-TamarilatticeassociahedrontropicalhyperplanearrangementpathspolytopalcomplexcanonicalrealizationHassediagram
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The pith

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

The reading

The paper constructs a geometric model for the alt ν-Tamari lattice on lattice paths lying weakly above a fixed path ν. It proves that the covering relations in this poset form the edges of a polytopal complex called the alt ν-associahedron, obtained from a tropical hyperplane arrangement. The construction uses a canonical coordinate system based on the areas under the paths. In the classical case this yields Loday's realization of the associahedron after an affine change of coordinates.

What carries the argument

The alt ν-associahedron, the polytopal complex whose 1-skeleton is the Hasse diagram of the alt ν-Tamari lattice and whose facets arise from a tropical hyperplane arrangement.

What would settle it

A single pair of lattice paths above ν whose covering relation in the Tamari order does not correspond to an edge in the tropical complex, or vice versa.

Watch

Extended reading notes

Core claim

The Hasse diagram of the alt ν-Tamari lattice coincides with the edge graph of the polytopal complex induced by a suitable tropical hyperplane arrangement; this complex is the alt ν-associahedron. Its vertices are the lattice paths above ν, and its canonical realization assigns to each path the vector of areas it subtends with the axes and with ν. When ν is the diagonal path the construction is affinely equivalent to Loday's classic realization of the associahedron.

Load-bearing premise

The covering relations of the alt ν-Tamari lattice exactly match the edges of the 1-skeleton coming from the tropical hyperplane arrangement.

Editorial extensions

If this is right

  • The same tropical arrangement supplies a geometric realization for every alt ν-Tamari lattice and for its special cases including the ν-Tamari and ν-Dyck lattices.
  • The canonical area-based coordinates give an explicit embedding that specializes to Loday's coordinates on the classical associahedron.
  • Combinatorial properties such as the number of facets or the diameter of the poset can be read from the geometry of the tropical arrangement.
  • The construction supplies a uniform polytopal model for a family of posets previously studied only combinatorially.

Reading between the lines

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

  • The area-based coordinates may extend to give realizations for other Tamari-like orders defined by different path constraints.
  • Because the construction recovers Loday's realization, it offers a bridge between tropical and classical combinatorial realizations of the associahedron.
  • One could test whether the same tropical hyperplanes produce higher associahedra or multi-associahedra when the path ν is replaced by a higher-dimensional analog.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 2 minor

Summary. The paper constructs a canonical realization of the alt ν-associahedron as the polytopal complex induced by a tropical hyperplane arrangement whose coordinates are the areas below lattice paths lying weakly above a fixed path ν. It claims that the 1-skeleton of this complex is exactly the Hasse diagram of the alt ν-Tamari lattice (generalizing the ν-associahedra of Ceballos–Padrol–Sarmiento) and that, for the classical case, the realization is affinely equivalent to Loday’s associahedron.

Significance. If the geometric identification holds, the construction supplies an explicit, area-based tropical model for a family of generalized Tamari posets. The parameter-free character of the area coordinates and the recovery of Loday’s realization are concrete strengths that would make the result useful for further combinatorial and geometric study.

major comments (1)
  1. [§3] §3 (main theorem on the 1-skeleton): the proof that the edges of the tropical complex coincide exactly with the covering relations of the alt ν-Tamari poset proceeds by showing that each combinatorial cover corresponds to a facet crossing, but does not contain an independent argument that the arrangement introduces no extraneous edges for arbitrary ν; the identification therefore inherits the poset structure from Ceballos–Chenevière rather than re-deriving it from the geometry of the hyperplanes.
minor comments (2)
  1. Notation for the area coordinates (e.g., the vector a(π)) is introduced without an explicit formula relating it to the standard height or inversion table; a displayed equation would improve readability.
  2. Figure 2 (classical case) would benefit from an explicit matrix or coordinate list showing the affine map to Loday’s realization.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. We address the single major comment below and will revise the manuscript to strengthen the proof.

read point-by-point responses

  1. Referee: [§3] §3 (main theorem on the 1-skeleton): the proof that the edges of the tropical complex coincide exactly with the covering relations of the alt ν-Tamari poset proceeds by showing that each combinatorial cover corresponds to a facet crossing, but does not contain an independent argument that the arrangement introduces no extraneous edges for arbitrary ν; the identification therefore inherits the poset structure from Ceballos–Chenevière rather than re-deriving it from the geometry of the hyperplanes.

    Authors: We acknowledge that the argument in Section 3 establishes one direction: every covering relation of the alt ν-Tamari poset corresponds to an edge by crossing exactly one facet. The converse—that the tropical arrangement introduces no extraneous edges—is not given an independent geometric proof and instead relies on the covering relations already established by Ceballos–Chenevière. We agree this is a substantive point. In the revised version we will add a self-contained argument showing that any two vertices joined by an edge in the arrangement differ by a covering relation; this will proceed by examining the area coordinates and proving that a non-covering difference forces the points to be separated by at least two hyperplanes. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; geometric realization is independent of prior combinatorial definitions

full rationale

The paper defines a new geometric model (tropical hyperplane arrangement with canonical area-below-path realization) and claims to show that its 1-skeleton matches the Hasse diagram of the alt ν-Tamari poset. This identification is presented as a derived result rather than an input or self-referential definition. Self-citations to Ceballos-Chenevière (poset) and Ceballos-Padrol-Sarmiento (prior realizations) supply the combinatorial starting point and context for generalization, but do not reduce the current construction or its claimed edge correspondence to a fit, renaming, or unverified self-citation chain. The recovery of Loday's realization via affine transform is an independent verification step. No equations or steps reduce the output to the input by construction.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The central claim rests on the prior combinatorial definition of the alt ν-Tamari lattice and on the geometric fact that a tropical hyperplane arrangement produces a polytopal complex whose 1-skeleton matches the Hasse diagram; no free parameters or new invented entities are visible in the abstract.

assumptions (1)
  • domain assumption The alt ν-Tamari lattice is a partial order on the set of lattice paths lying weakly above ν.
    Invoked in the first sentence of the abstract as the object whose Hasse diagram is realized.

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Cite this review

Pith. "Pith review of A canonical realization of the alt $\nu$-associahedron." pith.science (2026). https://pith.science/paper/UUOENENQ

@misc{pith2026240117204,
  author       = {Pith},
  title        = {Pith review of: A canonical realization of the alt $\nu$-associahedron},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UUOENENQ}},
  note         = {Machine review of arXiv:2401.17204}
}
abstract

Given a lattice path $\nu$, the alt $\nu$-Tamari lattice is a partial order recently introduced by Ceballos and Chenevi\`ere, which generalizes the $\nu$-Tamari lattice and the $\nu$-Dyck lattice. All these posets are defined on the set of lattice paths that lie weakly above $\nu$, and posses a rich combinatorial structure. In this paper, we study the geometric structure of these posets. We show that their Hasse diagram is the edge graph of a polytopal complex induced by a tropical hyperplane arrangement, which we call the alt $\nu$-associahedron. This generalizes the realization of $\nu$-associahedra by Ceballos, Padrol and Sarmiento. Our approach leads to an elegant construction, in terms of areas below lattice paths, which we call the canonical realization. Surprisingly, in the case of the classical associahedron, our canonical realization magically recovers Loday's ubiquitous realization, via a simple affine transformation.

Figures

Figures reproduced from arXiv: 2401.17204 by the authors.

Figure 1
Figure 1. Example of Loday’s coordinates of two plane binary trees. e1 e2 e3 e4 4321 4312 4213 3214 4141 4123 3124 2161 2134 1621 1612 1414 1261 1234 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Loday’s 3-dimensional associahedron. In this work, we unexpectedly discovered a new appearance of Loday’s associahedron (via a simple affine transformation) in a much wider generality. Before going into details about the general frame￾work, we would like to explain our construction in the special case of the classical associahedron to highlight the beauty of both constructions and their relation. Each plane binary t… view at source ↗
Figure 3
Figure 3. Example of coordinates of two plane binary trees in our canonical realization. x y z 343 243 133 022 323 123 012 301 001 340 240 020 300 000 [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (30 more)
Figure 4
Figure 4. Figure 4: Our canonical realization of the 3-dimensional associahedron. Magically, Loday’s associahedron and our canonical realization of the associahedron are related by the simple affine transformation ϕ : R n+1 → R n (2) ϕ(ℓ1, . . . , ℓn+1) = (cn, . . . , c1)(3) defined by (4…
Figure 5
Figure 5. Figure 5: The ν-Tamari lattice and ν-Dyck lattice for ν = ENEENN = (1, 2, 0, 0). They are the alt ν-Tamari lattices Tamν (δ) for δ = (2, 0, 0) and δ = (0, 0, 0), respec￾tively. 1200 03000210 0201 0111 0102 1110 0120 1101 0021 0012 1020 0030 10111002 0003 [PITH_FULL_IMAGE:figure…
Figure 6
Figure 6. Figure 6: The alt ν-Tamari lattice Tamν(δ) for ν = ENEENN = (1, 2, 0, 0) and δ = (1, 0, 0) [PITH_FULL_IMAGE:figures/full_fig_p004_6.png]
Figure 7
Figure 7. Figure 7: The δ-rotation of a ν-path for ν = (2, 2, 3, 1, 0) and δ = (1, 2, 0, 0). Each node is labelled with its δ-altitude. Definition 2.1. Let δ be an increment vector with respect to ν. The alt ν-Tamari poset Tamν (δ) is the reflexive transitive closure of δ-rotations on the…
Figure 8
Figure 8. Figure 8: Examples of alt ν-Tamari lattices Tamν(δ) for ν = ENEEN = (1, 2, 0). Left: the ν-Dyck lattice, for δ = (0, 0). Middle: the lattice for δ = (1, 0). Right: the ν-Tamari lattice, for δ = (2, 0). (2) The covering relations of Tamν(δ) are exactly δ-rotations. (3) For a fixe…
Figure 9
Figure 9. Figure 9: Left: lattice points in an a × b rectangle labeled by their position (i, j), for a = 8 and b = 3. Right: the stack set U of the sequence u = {1, 1, 2, 2, 4, 3, 3, 2, 1}. We say that two points p, q ∈ U are U-incompatible if p is strictly southwest or strictly northeast…
Figure 10
Figure 10. Figure 10: The left aligned stack set U for the sequence u = {4, 3, 3, 2, 2, 2, 1, 1, 1}. in U. We can visualize U-trees as classical rooted binary trees by connecting consecutive nodes in the same row or column. The vertex at the top-left corner of U is compatible with everyone…
Figure 11
Figure 11. Figure 11: The rotation of a U-tree. Let T be a U-tree and p, r ∈ T be two elements which do not lie in the same row or same column. We denote by pr the smallest rectangle containing p and r, and write pxr (resp. pqr) for the lower left corner (resp. upper right corner) of pr. L…
Figure 12
Figure 12. Figure 12: Examples of the rotation poset of U-trees for the stack sets U with unimodular sequences u = {2, 2, 3, 3}, u = {2, 3, 3, 2} and u = {3, 3, 2, 2}. In [10], Ceballos and Chenevi`ere showed that the alt ν-Tamari lattice Tamν(δ) can be described as a rotation poset of U-t…
Figure 13
Figure 13. Figure 13: Two examples of stack sets for ν = (0, 2, 3, 3). Left: the stack set U = Uδ,ν for δ = (2, 1, 1). Right: the stack set U ′ = Uδ ′ ,ν for δ ′ = (1, 2, 2). Theorem 2.7 ([10]). The alt ν-Tamari lattice Tamν(δ) and the rotation poset of (δ, ν)-trees are isomorphic: Tamν(δ)…
Figure 14
Figure 14. Figure 14: The right flushing bijection ϕδ,ν for ν = (2, 2, 3, 1, 0) and δ = (1, 2, 0, 0). Proposition 2.10 ([7, 10]). The right flushing ϕδ,ν is a bijection between the set of ν-paths and the set of (δ, ν)-trees. It sends δ-rotations on ν-paths to rotations on (δ, ν)-trees. Thu…
Figure 15
Figure 15. Figure 15: The classical Tamari lattice as the dual of the fine mixed subdivi￾sion M(U) for the stack set U of lattice points weakly above the path ν = ENEN. This polytope is subdivided into five cells corresponding to the five ν-trees, which are also shown in the figure. Their …
Figure 16
Figure 16. Figure 16: Minkowski sums of the cells in [PITH_FULL_IMAGE:figures/full_fig_p014_16.png]
Figure 17
Figure 17. Figure 17: Alternative realization of the classical Tamari lattice as the dual of the fine mixed subdivision M(U) for the stack set U of lattice points weakly above the path ν = NENENE. Recall that for x = (x0, x1, . . . , xn) with xi > 0 for all i, the Pitman–Stanley polytope i…
Figure 18
Figure 18. Figure 18: Examples of the fine mixed subdivisions M(Uδ,ν) of PUδ,ν for the path ν = EENEEN = (2, 2, 0) and the three possible choices of δ: (2, 0), (1, 0), and (0, 0). Therefore, for a fixed ν, the Hasse diagram of every alt ν-Tamari lattice Tamν(δ) can be realized as the dual …
Figure 19
Figure 19. Figure 19: An example of the duality between fine mixed subdivisions and arrange￾ments of (degenerate) tropical hyperplanes. Throughout the section, we fix a non-crossing height function h with respect to a unimodular shape U, and assume that the smallest rectangle containing U …
Figure 20
Figure 20. Figure 20: illustrates a single tropical hyperplane Hi intersected with the plane y0 = h(i, 0). If all h(i, j) are finite, then Hi subdivides the space TPd in d + 1 regions, where the minimum in Equa￾tion (9) is attained only once. In [PITH_FULL_IMAGE:figures/full_fig_p016_20.png]
Figure 21
Figure 21. Figure 21: Three arrangements Hh of tropical hyperplanes dual to the fine mixed subdivisions in [PITH_FULL_IMAGE:figures/full_fig_p017_21.png]
Figure 22
Figure 22. Figure 22: The set P + k (T ) is the set of nodes labeled + in the figure, while P − k (T ) is the set of nodes labeled −. The blue path is the unique path from the left most node of T in row k to the top of the shape. Proof. Let ℓ be the number of up vertical runs of the path f…
Figure 23
Figure 23. Figure 23: Area below the path Rk(T ). where Rk(T ) is the path that connects the left most node of T in row k to the root, and the area of a lattice path R is the number of boxes below it in the smallest rectangle containing it. Proof. The proof is a direct consequence of Propo…
Figure 24
Figure 24. Figure 24: The coordinate of the shown ν-tree T is g(T ) = (y1, y2, y3, y4) = (1, 6, 3, 5). Example 6.3 (The associahedron). The classical n-dimensional associahedron is obtained for ν = (EN) n. Our canonical realization for n = 2 is shown in [PITH_FULL_IMAGE:figures/full_fig_p…
Figure 25
Figure 25. Figure 25: The coordinates of the vertices of the canonical realization of the 2-dimensional associahedron are: (0, 0),(0, 1),(1, 2),(2, 2),(2, 0) [PITH_FULL_IMAGE:figures/full_fig_p020_25.png]
Figure 26
Figure 26. Figure 26: The 2-dimensional associahedron. Our canonical realization for n = 3 is shown in [PITH_FULL_IMAGE:figures/full_fig_p021_26.png]
Figure 27
Figure 27. Figure 27: The coordinates of the vertices of the canonical realization of the 3-dimensional associahedron [PITH_FULL_IMAGE:figures/full_fig_p021_27.png]
Figure 28
Figure 28. Figure 28: The 3-dimensional associahedron [PITH_FULL_IMAGE:figures/full_fig_p021_28.png]
Figure 29
Figure 29. Figure 29: The Fuss-Catalan associahedron for (m, n) = (2, 2) [PITH_FULL_IMAGE:figures/full_fig_p022_29.png]
Figure 30
Figure 30. Figure 30: The Fuss-Catalan associahedron for (m, n) = (3, 2). Example 6.5 (Two dimensional ν-associahedra). For a, b ∈ N and ν = EaNEbN, a schematic illustration of the ν-associahedron is shown in [PITH_FULL_IMAGE:figures/full_fig_p022_30.png]
Figure 31
Figure 31. Figure 31: Schematic figure of the 2-dimensional ν-associahedron for ν = E aNEbN. In this case, a = 4 and b = 5. Example 6.6 (Alt ν-associahedra for ν = ENEEN). For ν = ENEEN = (1, 2, 0), there are three possible choices for the increment vector δ: (2, 0),(1, 0) and (0, 0). The …
Figure 32
Figure 32. Figure 32: Alt ν associahedra for ν = (1, 2, 0) and three different choices of increment vector δ. The top is δ = (2, 0), the middle is δ = (1, 0), and the bottom is δ = (0, 0) [PITH_FULL_IMAGE:figures/full_fig_p024_32.png]
Figure 33
Figure 33. Figure 33: Alt ν associahedra for ν = (2, 2, 0) and two different choices of increment vector δ. The top is δ = (1, 0) and the bottom is δ = (0, 0). The case δ = (2, 0) is the Fuss-Catalan associahedron in [PITH_FULL_IMAGE:figures/full_fig_p025_33.png]

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Forward citations

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

  1. A combinatorial model for the canonical join complex of alt $\nu$-Tamari lattices

    math.CO 2026-05 unverdicted novelty 7.0 of 10

    The box complex realizes the canonical join complex of alt ν-Tamari lattices and is vertex decomposable, with Euler characteristic and top homology independent of δ.

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