pith. sign in

arxiv: 2401.17204 · v2 · pith:UUOENENQnew · submitted 2024-01-30 · 🧮 math.CO

A canonical realization of the alt ν-associahedron

Cesar Ceballos This is my paper

Pith reviewed 2026-05-24 04:20 UTC · model grok-4.3

classification 🧮 math.CO
keywords alt ν-Tamari latticeassociahedrontropical hyperplane arrangementlattice pathspolytopal complexcanonical realizationHasse diagram
0
0 comments X

The pith

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

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

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.

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.

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.

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.

Where Pith is reading between the lines

These are 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.

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.

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.

Figures

Figures reproduced from arXiv: 2401.17204 by Cesar Ceballos.

Figure 1
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. 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 plan… view at source ↗
Figure 3
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 ↗
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) ci = (ℓ1 + · · · + ℓi) − (1 + · · · + i). Our description of the canonical realization of the associahedron is just an example of a more g… view at source ↗
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:figures/full_fig_p004_5.png] view at source ↗
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] view at source ↗
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 set of ν-paths. Three examples of alt ν-Tamari lattices Tamν(δ) for ν = ENEEN = (1, 2, 0) and δ = (0, 0),(1, 0) and (2, 0) are illustrated … view at source ↗
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 fixed ν, all alt ν-Tamari lattices have the same number of linear intervals of any length.2 2.2. The U-Tamari lattice on trees. The proof of The… view at source ↗
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 of q, and all the lattice points in the smallest rectangle containing p and q belong to U. Otherwise, p and q are said to be U-compatible. … view at source ↗
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 else, and is the root of every U-tree. We often refer to the lattice points in a U-tree as nodes. An example of a U-tree and the rotation o… view at source ↗
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. Let p, q, r ∈ T be such that q = pxr and no other elements besides p, q, r lie in pr. The rotation of T at q is defined as the set T ′ = [PI… view at source ↗
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-trees for some specific stack set U. Our purpose now is to recall the construction of the set U in terms of δ and ν [PITH_FULL_IMAGE:figures… view at source ↗
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) ∼= Tam(Uδ,ν). The alt ν-Tamari lattices in [PITH_FULL_IMAGE:figures/full_fig_p008_13.png] view at source ↗
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. Thus, it is a poset isomorphism. The inverse of the right flushing bijection is called the left flushing bijection. It can be described similar… view at source ↗
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 Minkowski sums are shown in [PITH_FULL_IMAGE:figures/full_fig_p013_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Minkowski sums of the cells in [PITH_FULL_IMAGE:figures/full_fig_p014_16.png] view at source ↗
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 is defined as Πn+1(x) =  (y0, y1, . . . , yn) ∈ R n+1 : yi ≥ 0 and y0 + · · · + yi ≤ x0 + · · · + xi for 0 ≤ i ≤ n [PITH_FULL_IMAGE:figures… view at source ↗
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 graph of fine mixed subdivision of the same polytope Pν. Proof. Note that the polytope Pν = PUδmax,ν , where δ max = (ν1, . . . , νb). By Re… view at source ↗
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 is an a×b rectangle. We set h(i, j) = −∞, for every position (i, j) inside the a × b rectangle that is not inside U. The tropical projective… view at source ↗
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] view at source ↗
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] view at source ↗
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 from pk(T ) to p ′ k (T ) in T . These vertical runs alternate with ℓ − 1 left horizontal runs, see [PITH_FULL_IMAGE:figures/full_fig_p019_22.png] view at source ↗
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 Proposition 5.8 and is illustrated in [PITH_FULL_IMAGE:figures/full_fig_p020_23.png] view at source ↗
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_p020_24.png] view at source ↗
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] view at source ↗
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] view at source ↗
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] view at source ↗
Figure 28
Figure 28. Figure 28: The 3-dimensional associahedron [PITH_FULL_IMAGE:figures/full_fig_p021_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: The Fuss-Catalan associahedron for (m, n) = (2, 2) [PITH_FULL_IMAGE:figures/full_fig_p022_29.png] view at source ↗
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] view at source ↗
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 canonical realization of the corresponding alt ν-associahedra Assoδ,ν are illustrated in [PITH_FULL_IMAGE:figures/full_fig_p023_31.png] view at source ↗
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] view at source ↗
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] view at source ↗
read the original 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.

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

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

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.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 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.

axioms (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.

pith-pipeline@v0.9.0 · 5698 in / 1357 out tokens · 20446 ms · 2026-05-24T04:20:31.950392+00:00 · methodology

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.

Forward citations

Cited by 1 Pith paper

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

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

    math.CO 2026-05 unverdicted novelty 7.0

    A combinatorial model for the canonical join complex of alt ν-Tamari lattices proves vertex decomposability, shellability, and computes the homology.

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages · cited by 1 Pith paper

  1. [1]

    Tropical hyperplane a rrangements and oriented matroids

    Federico Ardila and Mike Develin. Tropical hyperplane a rrangements and oriented matroids. Math. Z. , 262(4):795– 816, 2009

    work page 2009

  2. [2]

    Higher trivariate diagonal harmonics via generalize d Tamari posets

    Fran¸ cois Bergeron and Louis-Fran¸ cois Pr´ eville-Ratelle. Higher trivariate diagonal harmonics via generalize d Tamari posets. J. Comb. , 3(3):317–341, 2012

    work page 2012

  3. [3]

    The representation of the sym- metric group on m-Tamari intervals

    Mireille Bousquet-M´ elou, Guillaume Chapuy, and Louis -Fran¸ cois Pr´ eville-Ratelle. The representation of the sym- metric group on m-Tamari intervals. Adv. Math. , 247:309–342, 2013

    work page 2013

  4. [4]

    The number of intervals in the m-Tamari lattices

    Mireille Bousquet-M´ elou, ´Eric Fusy, and Louis-Fran¸ cois Pr´ eville-Ratelle. The number of intervals in the m-Tamari lattices. Electron. J. Combin. , 18(2):Paper 31, 26, 2011

    work page 2011

  5. [5]

    Burkard, Bettina Klinz, and R¨ udiger Rudolf

    Rainer E. Burkard, Bettina Klinz, and R¨ udiger Rudolf. P erspectives of Monge properties in optimization. Discrete Appl. Math. , 70(2):95–161, 1996

    work page 1996

  6. [6]

    Geo metry of ν-Tamari lattices in types A and B

    Cesar Ceballos, Arnau Padrol, and Camilo Sarmiento. Geo metry of ν-Tamari lattices in types A and B. Trans. Amer. Math. Soc. , 371(4):2575–2622, 2019

    work page 2019

  7. [7]

    The ν-Tamari lattice via ν-trees, ν-bracket vectors, and subword complexes

    Cesar Ceballos, Arnau Padrol, and Camilo Sarmiento. The ν-Tamari lattice via ν-trees, ν-bracket vectors, and subword complexes. Electron. J. Combin. , 27(1):Paper No. 1.14, 31, 2020

    work page 2020

  8. [8]

    The s-weak order and s-permutahedra

    Cesar Ceballos and Viviane Pons. The s-weak order and s-permutahedra. S´ em. Lothar. Combin., 82B:Art. 76, 12, 2020

    work page 2020

  9. [9]

    Ziegle r

    Cesar Ceballos, Francisco Santos, and G¨ unter M. Ziegle r. Many non-equivalent realizations of the associahedron. Combinatorica, 35(5):513–551, 2015

    work page 2015

  10. [10]

    Chenevi` ere

    Ceballos. Cesar and Cl´ ement Chenevi` ere. On linear in tervals in the alt ν-tamari lattices. Preprint available at arXiv:2209.00418, May 2023. 24 CESAR CEBALLOS 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). A CANONICAL REALI...

    work page arXiv 2023

  11. [11]

    Chapoton

    F. Chapoton. Sur le nombre d’intervalles dans les treil lis de Tamari. S´ em. Lothar. Combin., 55:Art. B55f, 18, 2005/07

    work page 2005

  12. [12]

    De Loera, J¨ org Rambau, and Francisco Santos.Triangulations, volume 25 of Algorithms and Computation in Mathematics

    Jes´ us A. De Loera, J¨ org Rambau, and Francisco Santos.Triangulations, volume 25 of Algorithms and Computation in Mathematics . Springer-Verlag, Berlin, 2010. Structures for algorithm s and applications

    work page 2010

  13. [13]

    Tropical convexity

    Mike Develin and Bernd Sturmfels. Tropical convexity. Doc. Math. , 9:1–27, 2004

    work page 2004

  14. [14]

    Stiefel tropical linear s paces

    Alex Fink and Felipe Rinc´ on. Stiefel tropical linear s paces. J. Combin. Theory Ser. A , 135:291–331, 2015

    work page 2015

  15. [15]

    Constructing the associahedron

    Mark Haiman. Constructing the associahedron. Unpubli shed manuscript, MIT 1984, 11 pages, http://math.berkeley.edu/∼haiman/ftp/assoc/manuscript.pdf

    work page 1984

  16. [16]

    Mark D. Haiman. Conjectures on the quotient ring by diag onal invariants. J. Algebraic Combin. , 3(1):17–76, 1994

    work page 1994

  17. [17]

    The Cayley trick, lifting subdivisions and the Bohne-Dress theorem on zonotopal tilings

    Birkett Huber, J¨ org Rambau, and Francisco Santos. The Cayley trick, lifting subdivisions and the Bohne-Dress theorem on zonotopal tilings. J. Eur. Math. Soc. (JEMS) , 2(2):179–198, 2000

    work page 2000

  18. [18]

    Generalized triangulations and diagon al-free subsets of stack polyominoes

    Jakob Jonsson. Generalized triangulations and diagon al-free subsets of stack polyominoes. J. Combin. Theory Ser. A, 112(1):117–142, 2005

    work page 2005

  19. [19]

    Carl W. Lee. The associahedron and triangulations of th e n-gon. European J. Combin. , 10(6):551–560, 1989

    work page 1989

  20. [20]

    Realization of the Stasheff polytope

    Jean-Louis Loday. Realization of the Stasheff polytope . Arch. Math. (Basel) , 83(3):267–278, 2004

    work page 2004

  21. [21]

    Associahedra, Tamari lattices and related structures, volume 299 of Progress in Mathematics

    Folkert M¨ uller-Hoissen, Jean Marcel Pallo, and Jim St asheff, editors. Associahedra, Tamari lattices and related structures, volume 299 of Progress in Mathematics . Birkh¨ auser/Springer, Basel, 2012. Tamari memorial Festschrift

    work page 2012

  22. [22]

    Ziegl er

    Vincent Pilaud, Francisco Santos, and G¨ unter M. Ziegl er. Celebrating Loday’s associahedron. Arch. Math. (Basel) , 121(5-6):559–601, 2023

    work page 2023

  23. [23]

    Permutohedra, associahedra, an d beyond

    Alexander Postnikov. Permutohedra, associahedra, an d beyond. Int. Math. Res. Not. IMRN , (6):1026–1106, 2009

    work page 2009

  24. [24]

    The enumeration of generalized Tamari intervals

    Louis-Fran¸ cois Pr´ eville-Ratelle and Xavier Viennot. The enumeration of generalized Tamari intervals. Trans. Amer. Math. Soc. , 369(7):5219–5239, 2017

    work page 2017

  25. [25]

    The Cayley trick and triangulations of products of simplices

    Francisco Santos. The Cayley trick and triangulations of products of simplices. In Integer points in polyhedra— geometry, number theory, algebra, optimization , volume 374 of Contemp. Math. , pages 151–177. Amer. Math. Soc., Providence, RI, 2005

    work page 2005

  26. [26]

    Quantum groups , volume II of Graduate Texts in Mathematical Physics

    Steven Shnider and Shlomo Sternberg. Quantum groups , volume II of Graduate Texts in Mathematical Physics . International Press, Cambridge, MA, 1993. From coalgebras to Drinfel’d algebras, A guided tour

    work page 1993

  27. [27]

    Stanley and Jim Pitman

    Richard P. Stanley and Jim Pitman. A polytope related to empirical distributions, plane trees, parking functions, and the associahedron. Discrete Comput. Geom. , 27(4):603–634, 2002. TU Graz, Institut f ¨ur Geometrie, Kopernikusgasse 24, 8010 Graz, Austria. Email address : cesar.ceballos@tugraz.at

    work page 2002