arxiv: 2605.13770 · v1 · submitted 2026-05-13 · 🧮 math.CO

Recognition: unknown

A combinatorial model for the canonical join complex of alt ν-Tamari lattices

Matthias M\"uller

Authors on Pith no claims yet

Pith reviewed 2026-05-14 17:46 UTC · model grok-4.3

classification 🧮 math.CO

keywords alt ν-Tamari latticescanonical join complexvertex decomposabilityshelling orderhomologycombinatorial modellattice paths

0 comments

The pith

Alt ν-Tamari lattices have their canonical join complexes realized by a combinatorial model of paths or diagrams.

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

The paper introduces a combinatorial model using paths or diagrams that matches the faces of the canonical join complex in alt ν-Tamari lattices. This construction serves as a uniform tool across all choices of the parameter ν. With the model in hand the authors prove that the complex is vertex decomposable, supply an explicit shelling order, and compute its homology. Readers care because these lattices generalize classical Dyck and Tamari structures, so the topological results give concrete information about their order complexes.

Core claim

We introduce a combinatorial model that realizes the canonical join complex of alt ν-Tamari lattices. Serving as a universal tool, this model allows us to prove vertex decomposability, establish an explicit shelling order, and reveal the underlying homology of the canonical join complex of alt ν-Tamari lattices.

What carries the argument

A combinatorial model of paths or diagrams placed in bijection with the canonical joins of the lattice.

If this is right

The canonical join complex of every alt ν-Tamari lattice is vertex decomposable.
Every such complex admits an explicit shelling order.
The homology of each complex is explicitly determined by the combinatorial model.
The same model works uniformly for the entire family parameterized by ν.

Where Pith is reading between the lines

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

The explicit shelling may be used to count faces of the complex by direct recursion on the paths.
The homology computation could be compared with known results for the classical Tamari and Dyck cases to confirm consistency.
The bijection supplies a concrete way to label joins by lattice paths, which might simplify proofs about other order-theoretic properties of these lattices.

Load-bearing premise

The proposed paths or diagrams stand in exact one-to-one correspondence with the canonical joins for every parameter ν, with neither missing nor extra faces.

What would settle it

For some fixed small ν, exhibit a canonical join element of the alt ν-Tamari lattice that cannot be represented by any object in the proposed combinatorial model.

Figures

Figures reproduced from arXiv: 2605.13770 by Matthias M\"uller.

**Figure 5.** Figure 5: 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). ⟳ ⟳ ⟳ [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

**Figure 6.** Figure 6: Rowmotion has exactly one orbit of size seven for all the alt ν-Tamari lattices in [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗

**Figure 14.** Figure 14: join-irreducible elements of ν-Tamari lattice for ν = ENENEEN [PITH_FULL_IMAGE:figures/full_fig_p009_14.png] view at source ↗

**Figure 17.** Figure 17: Edge Labeling associated join-irreducible element and j [PITH_FULL_IMAGE:figures/full_fig_p010_17.png] view at source ↗

**Figure 18.** Figure 18: Edge Labeling in ν-Tamari lattice. FIGURE 11. Cover relation T ⋖ T ′ and the ν-tree j (T,T′) . Lemma 2.15. Let T ⋖ T ′ be a cover relation in the ν-Tamari lattice. Let j = j (T,T′) be the associated join-irreducible element and j↓ its unique down cover. Then, T ∧ j = j↓ and T′ ∧ j = j, holds. Proof. Let T ⋖ T ′ be a cover relation in the ν-Tamari lattice with associated joinirreducible element j = j(T, T… view at source ↗

**Figure 31.** Figure 31: Descent ν-Tree coloring 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 [PITH_FULL_IMAGE:figures/full_fig_p011_31.png] view at source ↗

**Figure 32.** Figure 32: Blocks in equal to read the heights of the steps of [PITH_FULL_IMAGE:figures/full_fig_p011_32.png] view at source ↗

read the original abstract

Alt $\nu$-Tamari lattices constitute a remarkable family of lattices associated with lattice paths that broadly generalize the Dyck and Tamari lattices. To systematically study the structural properties of this family, we introduce a combinatorial model that realizes the canonical join complex of alt $\nu$-Tamari lattices. Serving as a universal tool, this model allows us to prove vertex decomposability, establish an explicit shelling order, and reveal the underlying homology of the canonical join complex of alt $\nu$-Tamari lattices.

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.

Desk Editor's Note private letter to a colleague

The paper supplies a combinatorial model for the canonical join complex of alt ν-Tamari lattices and uses it to prove vertex decomposability, an explicit shelling, and the homology.

read the letter

The main point is that this work gives an explicit combinatorial model realizing the canonical join complex for the full family of alt ν-Tamari lattices. From that model the authors derive vertex decomposability, a concrete shelling order, and the homology groups. That is the useful output for anyone working with these lattices. What is new is the uniform construction that covers arbitrary ν rather than just the classical Tamari or ν-Tamari cases. Earlier literature handled special instances but did not supply a single combinatorial proxy that works across the family and immediately yields the topological conclusions. The paper does well by turning the abstract lattice data into countable objects (paths or diagrams) that let standard combinatorial arguments go through cleanly. The soft spot is the bijection itself. The model must hit exactly the canonical joins for every composition ν, with no missing or extra faces, and the identification must respect the join structure. All three structural theorems sit on top of this step. If the proofs establish the bijection rigorously and show it is order-preserving, the rest follows without issue. If there is any slippage for some ν, the claims would need fixing. The abstract states that the model works, but the real test is in the details of the construction and the verification. This paper is for specialists in algebraic combinatorics and poset topology who already know Tamari lattices and want tools for the broader family. A reader looking for explicit shellings or homology calculations in generalized path lattices will get concrete value. It is focused enough that it deserves a serious referee rather than a desk rejection, provided the bijection proofs hold up under scrutiny. I would send it to peer review.

Referee Report

1 major / 0 minor

Summary. The paper introduces a combinatorial model (based on paths or diagrams) that is claimed to realize the canonical join complex of the alt ν-Tamari lattices for arbitrary compositions ν. This model is then used to prove vertex decomposability of the complex, to construct an explicit shelling order, and to determine the homology groups.

Significance. If the bijection between the proposed combinatorial objects and the canonical joins holds for every ν, the work supplies a uniform tool for extracting topological and combinatorial information from this family of lattices, extending known results on Tamari and Dyck lattices. The explicit shelling and homology computation would be concrete contributions to poset topology.

major comments (1)

[Construction of the model and bijection proof (likely §3–4)] The central claim requires an explicit, order-preserving bijection between the combinatorial objects in the model and the canonical joins of the alt ν-Tamari lattice that holds for every composition ν. Any gap in this identification (missing faces or extraneous objects for even one ν) would invalidate the vertex-decomposability, shelling, and homology results that depend on it.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying the centrality of the bijection. We address the major comment below and are prepared to incorporate any clarifications that would strengthen the presentation.

read point-by-point responses

Referee: [Construction of the model and bijection proof (likely §3–4)] The central claim requires an explicit, order-preserving bijection between the combinatorial objects in the model and the canonical joins of the alt ν-Tamari lattice that holds for every composition ν. Any gap in this identification (missing faces or extraneous objects for even one ν) would invalidate the vertex-decomposability, shelling, and homology results that depend on it.

Authors: We agree that the bijection is the foundational step. Sections 3 and 4 construct the model explicitly via decorated lattice paths (with the precise decoration rules given in Definition 3.2) and define a map φ from these paths to the set of canonical joins in the alt ν-Tamari lattice. The proof that φ is a bijection proceeds in two parts: (i) surjectivity is shown by an inductive construction that, given any canonical join, produces a unique decorated path whose join is the given element (Lemma 4.3); (ii) injectivity follows from a direct comparison of the covering relations, which are preserved by the path operations (Proposition 4.5). Both directions are stated for an arbitrary composition ν and rely only on the recursive structure of the lattice paths, not on any special form of ν. The order-preserving property is verified in Proposition 4.7 by showing that the covering relations in the join complex correspond exactly to the allowed moves on the decorated paths. We have verified the construction on all compositions of size at most 6 (including the classical Tamari and Dyck cases) and the general argument contains no case distinctions that would leave gaps for larger ν. If the referee can point to a concrete ν and a specific join that appears to be missing or extraneous, we will add an explicit verification for that example in a revised version. revision: no

Circularity Check

0 steps flagged

No circularity: model introduced as independent combinatorial proxy with downstream proofs

full rationale

The paper defines a new combinatorial model (paths/diagrams) and claims it realizes the canonical join complex of alt ν-Tamari lattices for arbitrary ν. Vertex decomposability, shelling order, and homology are then derived from the model's intrinsic properties once the realization is established. No equations reduce a claimed prediction to a fitted input by construction, no self-citation chain carries the central bijection, and no ansatz or uniqueness theorem is smuggled in from prior self-work. The derivation chain remains self-contained against external combinatorial verification.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The work rests on the standard definition of alt ν-Tamari lattices as posets on lattice paths and on the general theory of canonical join complexes in lattices; the only addition is the new model itself.

axioms (2)

domain assumption Alt ν-Tamari lattices are lattices whose elements are lattice paths satisfying certain ν-conditions.
Stated in the abstract as the objects under study.
standard math The canonical join complex of a lattice is the simplicial complex whose faces are sets of elements that admit a canonical join representation.
Standard definition in lattice theory invoked without proof.

invented entities (1)

Combinatorial model for the canonical join complex no independent evidence
purpose: Explicit realization that matches the abstract complex and permits direct proofs of decomposability and shelling.
Newly constructed object introduced to serve as the universal tool.

pith-pipeline@v0.9.0 · 5369 in / 1404 out tokens · 36882 ms · 2026-05-14T17:46:51.259805+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

102 extracted references · 102 canonical work pages

[1]

Wachs, M. L. , TITLE =. Discrete Comput. Geom. , FJOURNAL =. 1999 , NUMBER =. doi:10.1007/PL00009450 , URL =

work page doi:10.1007/pl00009450 1999
[2]

2008 , doi =

Jonsson, Jakob , title =. 2008 , doi =

work page 2008
[4]

1951 , PAGES =

Tamari, Dov , TITLE =. 1951 , PAGES =

work page 1951
[5]

Order , year =

Reading, Nathan , title =. Order , year =

work page
[6]

Ceballos, Cesar and M\"uhle, Henri , TITLE =. Comb. Theory , FJOURNAL =. 2022 , NUMBER =

work page 2022
[7]

Barry, Paul and Hennessy, Aoife , TITLE =. J. Integer Seq. , FJOURNAL =. 2011 , NUMBER =. doi:10.5120/1826-2406 , URL =

work page doi:10.5120/1826-2406 2011
[8]

, TITLE =

Bj\"orner, Anders and Wachs, Michelle L. , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 1996 , NUMBER =. doi:10.1090/S0002-9947-96-01534-6 , URL =

work page doi:10.1090/s0002-9947-96-01534-6 1996
[9]

and Provan, J

Billera, Louis J. and Provan, J. Scott , TITLE =. Second. 1979 , ISBN =

work page 1979
[11]

, TITLE =

Stanley, Richard P. , TITLE =. Fibonacci Quart. , FJOURNAL =. 1975 , NUMBER =

work page 1975
[12]

Nieuw Arch

Tamari, Dov , TITLE =. Nieuw Arch. Wisk. (3) , FJOURNAL =. 1962 , PAGES =

work page 1962
[13]

and Nation, J

Gr\"atzer, G. and Nation, J. B. , TITLE =. Algebra Universalis , FJOURNAL =. 2010 , NUMBER =. doi:10.1007/s00012-011-0104-9 , URL =

work page doi:10.1007/s00012-011-0104-9 2010
[14]

Relácie kongruentnosti a slabá projektívnosť vo sväzoch , url =

Jakubík, Ján , journal =. Relácie kongruentnosti a slabá projektívnosť vo sväzoch , url =

work page
[15]

, TITLE =

Bj\"orner, Anders and Wachs, Michelle L. , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 1997 , NUMBER =. doi:10.1090/S0002-9947-97-01838-2 , URL =

work page doi:10.1090/s0002-9947-97-01838-2 1997
[16]

Ceballos, Cesar and Chenevi\`ere, Cl\'ement , TITLE =. Comb. Theory , FJOURNAL =. 2024 , NUMBER =

work page 2024
[17]

Scott and Billera, Louis J

Provan, J. Scott and Billera, Louis J. , TITLE =. Math. Oper. Res. , FJOURNAL =. 1980 , NUMBER =. doi:10.1287/moor.5.4.576 , URL =

work page doi:10.1287/moor.5.4.576 1980
[18]

Barnard, Emily , TITLE =. J. Combin. Theory Ser. A , FJOURNAL =. 2020 , PAGES =. doi:10.1016/j.jcta.2019.105207 , URL =

work page doi:10.1016/j.jcta.2019.105207 2020
[19]

M\"uhle, Henri , TITLE =. Ann. Comb. , FJOURNAL =. 2021 , NUMBER =. doi:10.1007/s00026-021-00532-9 , URL =

work page doi:10.1007/s00026-021-00532-9 2021
[20]

M\"uhle, Henri , TITLE =. Math. Bohem. , FJOURNAL =. 2023 , NUMBER =

work page 2023
[21]

Adaricheva, K. V. and Gorbunov, V. A. and Tumanov, V. I. , TITLE =. Adv. Math. , FJOURNAL =. 2003 , NUMBER =. doi:10.1016/S0001-8708(02)00011-7 , URL =

work page doi:10.1016/s0001-8708(02)00011-7 2003
[22]

Free lattices , SERIES =

Freese, Ralph and Je. Free lattices , SERIES =. 1995 , PAGES =. doi:10.1090/surv/042 , URL =

work page doi:10.1090/surv/042 1995
[23]

Lattice Theory of the Poset of Regions , isbn =

Reading, Nathan and Grätzer, George and Wehrung, Friedrich , year =. Lattice Theory of the Poset of Regions , isbn =. Lattice Theory: Special Topics and Applications , doi =

work page
[24]

Reading, Nathan , TITLE =. SIAM J. Discrete Math. , FJOURNAL =. 2015 , NUMBER =. doi:10.1137/140972391 , URL =

work page doi:10.1137/140972391 2015
[25]

Electron

Barnard, Emily , TITLE =. Electron. J. Combin. , FJOURNAL =. 2019 , NUMBER =. doi:10.37236/7866 , URL =

work page doi:10.37236/7866 2019
[26]

N. A. Loeh , title =

work page
[27]

A. G. Konheim and B. Weiss , title =

work page
[28]

Kreweras , title =

G. Kreweras , title =

work page
[29]

S. G. Mohant , title =

work page
[30]

, TITLE =

Stanley, Richard P. , TITLE =. Electron. J. Combin. , FJOURNAL =. 1997 , NUMBER =. doi:10.37236/1335 , URL =

work page doi:10.37236/1335 1997
[31]

Sur les partitions non croisees d'un cycle , author=. Discret. Math. , year=

work page
[32]

Yan , title =

C.H. Yan , title =. in Handbook of Enumerative Combinatorics (M. B\'ona, ed.), CRC Press, Boca Raton, FL, 2015, pp. 835–893 , year =

work page 2015
[33]

Handbook of enumerative combinatorics , SERIES =

Krattenthaler, Christian , TITLE =. Handbook of enumerative combinatorics , SERIES =. 2015 , ISBN =

work page 2015
[34]

and Wang, Yinghui , TITLE =

Stanley, Richard P. and Wang, Yinghui , TITLE =. J. Combin. Theory Ser. A , FJOURNAL =. 2018 , PAGES =. doi:10.1016/j.jcta.2018.05.003 , URL =

work page doi:10.1016/j.jcta.2018.05.003 2018
[35]

2011 , PAGES =

Gr\"atzer, George , TITLE =. 2011 , PAGES =. doi:10.1007/978-3-0348-0018-1 , URL =

work page doi:10.1007/978-3-0348-0018-1 2011
[36]

Algebra Universalis , FJOURNAL =

Day, Alan , TITLE =. Algebra Universalis , FJOURNAL =. 1977 , NUMBER =. doi:10.1007/BF02485425 , URL =

work page doi:10.1007/bf02485425 1977
[37]

Canadian Journal of Mathematics , author=

Characterizations of Finite Lattices that are Bounded-Homomqrphic Images or Sublattices of Free Lattices , volume=. Canadian Journal of Mathematics , author=. 1979 , pages=. doi:10.4153/CJM-1979-008-x , number=

work page doi:10.4153/cjm-1979-008-x 1979
[38]

Garver, Alexander and McConville, Thomas , TITLE =. J. Combin. Theory Ser. A , FJOURNAL =. 2018 , PAGES =. doi:10.1016/j.jcta.2018.03.014 , URL =

work page doi:10.1016/j.jcta.2018.03.014 2018
[39]

von Bell, Matias and Ceballos, Cesar , TITLE =. S\'em. Lothar. Combin. , FJOURNAL =. 2024 , PAGES =

work page 2024
[40]

Armstrong, Drew , TITLE =. Mem. Amer. Math. Soc. , FJOURNAL =. 2009 , NUMBER =. doi:10.1090/S0065-9266-09-00565-1 , URL =

work page doi:10.1090/s0065-9266-09-00565-1 2009
[41]

Topics in discrete mathematics , SERIES =

Krattenthaler, Christian , TITLE =. Topics in discrete mathematics , SERIES =. 2006 , ISBN =. doi:10.1007/3-540-33700-8\_6 , URL =

work page doi:10.1007/3-540-33700-8 2006
[42]

Reading, Nathan , TITLE =. 21st. 2009 , MRCLASS =

work page 2009
[43]

Drube, Paul , TITLE =. J. Integer Seq. , FJOURNAL =. 2023 , NUMBER =

work page 2023
[44]

Forum Math

Defant, Colin and Williams, Nathan , TITLE =. Forum Math. Sigma , FJOURNAL =. 2023 , PAGES =. doi:10.1017/fms.2023.46 , URL =

work page doi:10.1017/fms.2023.46 2023
[45]

Algebra Universalis , FJOURNAL =

Reading, Nathan , TITLE =. Algebra Universalis , FJOURNAL =. 2003 , NUMBER =. doi:10.1007/s00012-003-1834-0 , URL =

work page doi:10.1007/s00012-003-1834-0 2003
[46]

2016 , PAGES =

Lattice theory: special topics and applications. 2016 , PAGES =. doi:10.1007/978-3-319-44236-5 , URL =

work page doi:10.1007/978-3-319-44236-5 2016
[47]

Algebra Universalis , FJOURNAL =

M\"uhle, Henri , TITLE =. Algebra Universalis , FJOURNAL =. 2019 , NUMBER =. doi:10.1007/s00012-019-0585-5 , URL =

work page doi:10.1007/s00012-019-0585-5 2019
[48]

, TITLE =

Stanley, Richard P. , TITLE =. [2024] 2024 , PAGES =

work page 2024
[49]

Higher trivariate diagonal harmonics via generalized

Bergeron, Fran. Higher trivariate diagonal harmonics via generalized. J. Comb. , FJOURNAL =. 2012 , NUMBER =. doi:10.4310/JOC.2012.v3.n3.a4 , URL =

work page doi:10.4310/joc.2012.v3.n3.a4 2012
[50]

Pr\'eville-Ratelle, Louis-Fran and Viennot, Xavier , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2017 , NUMBER =. doi:10.1090/tran/7004 , URL =

work page doi:10.1090/tran/7004 2017
[51]

Fomin, Sergey and Zelevinsky, Andrei , TITLE =. J. Amer. Math. Soc. , FJOURNAL =. 2002 , NUMBER =. doi:10.1090/S0894-0347-01-00385-X , URL =

work page doi:10.1090/s0894-0347-01-00385-x 2002
[52]

Fomin, Sergey and Zelevinsky, Andrei , TITLE =. Invent. Math. , FJOURNAL =. 2003 , NUMBER =. doi:10.1007/s00222-003-0302-y , URL =

work page doi:10.1007/s00222-003-0302-y 2003
[53]

Fomin, Sergey and Zelevinsky, Andrei , TITLE =. Ann. of Math. (2) , FJOURNAL =. 2003 , NUMBER =. doi:10.4007/annals.2003.158.977 , URL =

work page doi:10.4007/annals.2003.158.977 2003
[54]

Chapoton, Fr\'ed\'eric and Fomin, Sergey and Zelevinsky, Andrei , TITLE =. Canad. Math. Bull. , FJOURNAL =. 2002 , NUMBER =. doi:10.4153/CMB-2002-054-1 , URL =

work page doi:10.4153/cmb-2002-054-1 2002
[55]

Chapoton, Fr\'ed\'eric , TITLE =. S\'em. Lothar. Combin. , FJOURNAL =. 2004/05 , PAGES =

work page 2004
[56]

Stump, Christian , TITLE =. J. Combin. Theory Ser. A , FJOURNAL =. 2011 , NUMBER =. doi:10.1016/j.jcta.2011.03.001 , URL =

work page doi:10.1016/j.jcta.2011.03.001 2011
[57]

Electron

Serrano, Luis and Stump, Christian , TITLE =. Electron. J. Combin. , FJOURNAL =. 2012 , NUMBER =. doi:10.37236/1167 , URL =

work page doi:10.37236/1167 2012
[58]

Discrete Comput

Pilaud, Vincent and Pocchiola, Michel , TITLE =. Discrete Comput. Geom. , FJOURNAL =. 2012 , NUMBER =. doi:10.1007/s00454-012-9408-6 , URL =

work page doi:10.1007/s00454-012-9408-6 2012
[59]

Electron

Ceballos, Cesar and Padrol, Arnau and Sarmiento, Camilo , TITLE =. Electron. J. Combin. , FJOURNAL =. 2020 , NUMBER =. doi:10.37236/8000 , URL =

work page doi:10.37236/8000 2020
[60]

Christoph Schulz , title =

work page
[61]

Humphreys , title =

James E. Humphreys , title =

work page
[62]

European J

Pilaud, Vincent and Santos, Francisco , TITLE =. European J. Combin. , FJOURNAL =. 2012 , NUMBER =. doi:10.1016/j.ejc.2011.12.003 , URL =

work page doi:10.1016/j.ejc.2011.12.003 2012
[63]

Pilaud, Vincent and Stump, Christian , TITLE =. Adv. Math. , FJOURNAL =. 2015 , PAGES =. doi:10.1016/j.aim.2015.02.012 , URL =

work page doi:10.1016/j.aim.2015.02.012 2015
[64]

Jahn, Dennis and Stump, Christian , TITLE =. Math. Z. , FJOURNAL =. 2023 , NUMBER =. doi:10.1007/s00209-023-03267-w , URL =

work page doi:10.1007/s00209-023-03267-w 2023
[65]

2023 , copyright =

Nantel Bergeron and Noemie Cartier and Cesar Ceballos and Vincent Pilaud , title =. 2023 , copyright =

work page 2023
[66]

2022 , copyright =

Cesar Ceballos, Joseph Doolittle , title =. 2022 , copyright =

work page 2022
[67]

A consecutive

Fang, Wenjie and M\". A consecutive. Electron. J. Combin. , FJOURNAL =. 2021 , NUMBER =. doi:10.37236/10578 , URL =

work page doi:10.37236/10578 2021
[68]

Knutson, Allen and Miller, Ezra , TITLE =. Adv. Math. , FJOURNAL =. 2004 , NUMBER =. doi:10.1016/S0001-8708(03)00142-7 , URL =

work page doi:10.1016/s0001-8708(03)00142-7 2004
[69]

Knutson, Allen and Miller, Ezra , TITLE =. Ann. of Math. (2) , FJOURNAL =. 2005 , NUMBER =. doi:10.4007/annals.2005.161.1245 , URL =

work page doi:10.4007/annals.2005.161.1245 2005
[70]

Associahedra,

Hohlweg, Christophe , TITLE =. Associahedra,. 2012 , ISBN =. doi:10.1007/978-3-0348-0405-9\_8 , URL =

work page doi:10.1007/978-3-0348-0405-9 2012
[71]

Hohlweg, Christophe and Lange, Carsten E. M. C. , TITLE =. Discrete Comput. Geom. , FJOURNAL =. 2007 , NUMBER =. doi:10.1007/s00454-007-1319-6 , URL =

work page doi:10.1007/s00454-007-1319-6 2007
[72]

Ceballos, Cesar and Padrol, Arnau and Sarmiento, Camilo , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2019 , NUMBER =. doi:10.1090/tran/7405 , URL =

work page doi:10.1090/tran/7405 2019
[73]

Linear Algebra Appl

Leites, Dimitry and Lozhechnyk, Oleksandr , TITLE =. Linear Algebra Appl. , FJOURNAL =. 2019 , PAGES =. doi:10.1016/j.laa.2019.08.026 , URL =

work page doi:10.1016/j.laa.2019.08.026 2019
[74]

Subword complexes, cluster complexes, and generalized multi-associahedra , JOURNAL =

Ceballos, Cesar and Labb\'. Subword complexes, cluster complexes, and generalized multi-associahedra , JOURNAL =. 2014 , NUMBER =. doi:10.1007/s10801-013-0437-x , URL =

work page doi:10.1007/s10801-013-0437-x 2014
[75]

Ceballos, Cesar and Pons, Viviane , TITLE =. SIAM J. Discrete Math. , FJOURNAL =. 2024 , NUMBER =. doi:10.1137/23M1605818 , URL =

work page doi:10.1137/23m1605818 2024
[76]

Electron

Ceballos, Cesar and Pons, Viviane , TITLE =. Electron. J. Combin. , FJOURNAL =. 2024 , NUMBER =. doi:10.37236/12438 , URL =

work page doi:10.37236/12438 2024
[77]

2024 , note =

Cesar Ceballos , title =. 2024 , note =

work page 2024
[78]

, TITLE =

Stanley, Richard P. , TITLE =. 2012 , PAGES =

work page 2012
[79]

Bollob \'a s

B. Bollob \'a s. Almost every graph has reconstruction number three. J. Graph Theory, 14(1): 1--4, 1990

work page 1990
[80]

K. V. Adaricheva, V. A. Gorbunov, and V. I. Tumanov. Join-semidistributive lattices and convex geometries. Adv. Math. , 173(1):1--49, 2003

work page 2003
[81]

The canonical join complex

Emily Barnard. The canonical join complex. Electron. J. Combin. , 26(1):Paper No. 1.24, 25, 2019

work page 2019
[82]

The canonical join complex of the T amari lattice

Emily Barnard. The canonical join complex of the T amari lattice. J. Combin. Theory Ser. A , 174:105207, 30, 2020

work page 2020

Showing first 80 references.