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arxiv: 2607.01206 · v1 · pith:KFCEJNHBnew · submitted 2026-07-01 · 🧮 math.CO

Decomposition of Greedy Tamari Intervals and Bipartite Planar Maps

Philippe Biane , Wenjie Fang This is my paper

Pith reviewed 2026-07-02 09:54 UTC · model grok-4.3

classification 🧮 math.CO
keywords greedy Tamari posetbipartite planar mapsrecursive decompositioncombinatorial enumerationTamari intervalsposet intervalsplanar maps
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The pith

Greedy Tamari intervals decompose recursively exactly as bipartite planar maps do.

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

The paper shows that intervals in the greedy Tamari poset admit a recursive decomposition that is isomorphic to the standard recursive decomposition of bipartite planar maps. This isomorphism supplies a direct combinatorial proof that the two families are equi-enumerous for the m=1 case and simultaneously produces the refined enumeration that had been conjectured earlier. A reader cares because the argument replaces an algebraic functional equation with an explicit, structure-preserving bijection between two well-studied combinatorial objects.

Core claim

We give a combinatorial proof for the case m=1 that intervals of the greedy Tamari poset are equi-enumerous to bipartite planar maps by establishing a recursive decomposition of greedy Tamari intervals isomorphic to that of bipartite planar maps. This decomposition also yields the refined enumeration conjectured by Bousquet-Mélou and Chapoton. A more general refined conjecture is proposed for arbitrary m.

What carries the argument

The recursive decomposition of greedy Tamari intervals, shown to be isomorphic to the standard decomposition of bipartite planar maps.

If this is right

  • The total number of greedy Tamari intervals equals the number of bipartite planar maps.
  • A refined count by additional parameters (such as size and another statistic) matches the conjectured formula.
  • The same decomposition technique supplies an explicit bijection between the two families.
  • The approach suggests a route to refined enumeration for the general greedy m-Tamari case via a similar isomorphism.

Where Pith is reading between the lines

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

  • The isomorphism may extend to other families of Tamari-like posets or to constellations for m>1.
  • The decomposition could be used to transfer algorithms or generating-function techniques from planar maps back to Tamari intervals.
  • It raises the question whether similar structure-preserving decompositions exist between greedy nu-Tamari intervals and higher-constellation maps.

Load-bearing premise

The recursive decomposition defined on greedy Tamari intervals is isomorphic to the standard decomposition used for bipartite planar maps.

What would settle it

An explicit count of greedy Tamari intervals of a given size that differs from the known enumeration of bipartite planar maps of the corresponding size, or a concrete interval whose decomposition tree cannot be matched to any bipartite map decomposition.

Figures

Figures reproduced from arXiv: 2607.01206 by Philippe Biane, Wenjie Fang.

Figure 1
Figure 1. Figure 1: Example of the cover relation ⋖g, and the greedy Tamari poset (D4, ≤g) of size 4 wT 1 3 2 1 1 1 2 1 1 asc(T) = (1, 3, 2, 1, 1, 1, 2, 1, 1) u1 u0 u2 u3 u4 u5 u6 u7 u8 u9 u10u11 u12 u13 T Ascents of T [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Example of a plane tree T, its ascent profile asc(T) and its contour walk wT Note that, if in the previous description we take w ◦ to be the shortest prefix, then we get the cover relation for the classical Tamari order. This implies that the greedy Tamari order is coarser that the classical order. 2.2. Plane trees and Dyck paths. We denote by Tn the set of rooted plane trees with n edges, which are always… view at source ↗
Figure 3
Figure 3. Figure 3: Example of slide-up operation and the poset (T4, ≤g). T partition edges e1, . . . , en of T into parts whose edge indices form intervals. The ascent profile of T, denoted by asc(T), is the sequence of lengths (a1, . . . , ar) of ascents in the left-to-right contour order. We note that fasc(T) = a1 in this case. See [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Example of a (rooted) bipartite planar map corner c, which form the part of T which is reattached to obtain T ′ in the slide-up at v ′′ . As the effect of sliding up on the contour path is exactly exchanging w ◦ with the 0 that precedes it, which corresponds to the second visit to the edge (v, v”), we have the full equivalence. □ Using Proposition 2.4, we will take (Tn, ≤g) as the greedy Tamari poset herei… view at source ↗
Figure 5
Figure 5. Figure 5: Decomposition of bipartite planar maps Here, Ω is a linear operator in the space of polynomials in x, extended naturally to the space of power series in t with coefficients polynomial in x, defined by (5) ∀i ≥ 0, Ωx i = X i k=1 x i−k pk. Equation (4), given in [Tut62] by the name of “even slicing”, can be explained combina￾torially as follows. Taking out the edge next to the root in clockwise order of a no… view at source ↗
Figure 6
Figure 6. Figure 6: The only non-trivial commutation of slide-up operations for cover relations in the greedy Tamari poset • The node ui is the next sibling of the parent of uj . In this case, as illustrated in [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Example of the midpoint tree of a greedy Tamari interval I = (T, T′ ), with backbone and side forest distinguished by colors have also studied other aspects of the order structure of the greedy Tamari poset, which will be detailed in a forthcoming article. Given an interval I ∈ In, we define the canonical chain of I to be the chain described in Corollary 3.6. In view of Lemma 3.4 and Corollary 3.6, we give… view at source ↗
Figure 8
Figure 8. Figure 8: Example of the first construction ∆(I, k) ≤g head body I = (T, T′ ) [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: An example of the head and body of T in a greedy Tamari interval I = (T, T′ ) The second construction will produce an interval I = (S, S′ ) from two smaller intervals I1 = (T1, T′ 1 ) and I2 = (T2, T′ 2 ). Roughly, S ′ will be simply obtained by putting T ′ 2 above T ′ 1 , and for S we will need to cut T1 into two parts and insert T2 inbetween. We now detail the precise construction. Given I = (T, T′ ) ∈ I… view at source ↗
Figure 10
Figure 10. Figure 10: An example of the construction ⊕(I1, I2) illustrated in [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Example of the decomposition of a type-2 greedy Tamari interval By the expression of ci(Smid) and the construction of T2, with the change of variable j ′ = i ′ + j∗ + 1, this condition is equivalent to j X′−h i=j∗−h+1 ci(Smid) ≥ j X′ i=j∗+1 ci(S ′ ) for all j∗ < j′ ≤ k − 2. If we further suppose (C2) and (C3), hence Equation (6), we have (7) j X′−h i=0 ci(Smid) ≥ j X′ i=0 ci(S ′ ) for all j∗ < j′ ≤ k − 2.… view at source ↗
Figure 12
Figure 12. Figure 12: Example of a covering relation in the greedy 2-Tamari poset of order 9, in both forms of 2-ballot paths and 2-branch trees 4. Greedy m-Tamari intervals and (m + 1)-constellations Definition 4.1. Given m ≥ 1, an m-ballot path is a Dyck path such that the lengths of its ascents are multiples of m. The length of the Dyck path is a multiple of m, say mn and we call n its size and denote by D (m) n ⊆ Dmn the s… view at source ↗
Figure 13
Figure 13. Figure 13: Greedy 2-Tamari poset of order 3, with 2-ballot paths as elements on the left, and with m-branch trees as elements on the right. where both asc(I) = (a1, . . . , ar) and fasc(I) are taken for I as a greedy Tamari interval in (Tmn, ≤g). The generating function is thus (10) FI (m) (t, x) = X n≥0 t n X I∈I(m) n wtm(I). 4.1. Planar constellations and their decomposition. A planar m-constellation is a special … view at source ↗
Figure 14
Figure 14. Figure 14: Example of an m-constellation define FC (m) (t, x) ≡ FC (m) (t, x; p1, p2, . . .) by (11) FC (m) (t, x) = X n≥0 t n X C∈C(m) n x outdeg(C) Y f inner face of C pdeg(f) . In [Fan16, Section 4.1], a recursive decomposition of planar m-constellations is given, leading to the following functional equation: (12) FC (m) (t, x) = 1 + tx(FC (m) + Ω)m(1). 4.2. Refined and generalized conjecture on greedy m-Tamari i… view at source ↗
read the original abstract

The greedy Tamari poset, inspired by the well-studied Tamari lattice, was recently defined by Dermenjian in the more general setting of greedy $\nu$-Tamari posets. Bousquet-M\'elou and Chapoton counted intervals of the greedy $m$-Tamari poset in 2024 by solving a functional equation, and found that they are equi-enumerous to planar $(m+1)$-constellations. In this work, we give a combinatorial proof of this fact for the case $m = 1$, which also gives the refined enumeration conjectured by Bousquet-M\'elou and Chapoton. This is done by establishing a recursive decomposition of greedy Tamari intervals isomorphic to that of bipartite planar maps. We also propose a more general and refined conjecture for the case of general $m$.

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 / 0 minor

Summary. The paper claims to give a combinatorial proof that the number of intervals in the greedy 1-Tamari poset equals the number of bipartite planar maps (and a refined enumeration conjectured by Bousquet-Mélou and Chapoton) by constructing an explicit recursive decomposition of greedy Tamari intervals that is isomorphic to the standard decomposition of bipartite planar maps; it also states a more general conjecture for m>1.

Significance. If the claimed isomorphism between the two recursive decompositions holds at the level of unique decompositions and refined statistics, the result supplies the first direct combinatorial correspondence between these objects and resolves the refined enumeration conjecture for m=1.

major comments (1)
  1. [Abstract, paragraph 3 (and the decomposition section that follows)] The central claim rests on the assertion (abstract, paragraph 3) that the recursive decomposition defined for greedy Tamari intervals is isomorphic to the standard decomposition of bipartite planar maps, including preservation of the refined enumeration parameters. The manuscript must supply an explicit, case-by-case verification that every interval decomposes uniquely into a root component plus a sequence of smaller intervals whose sizes and labels match the map decomposition exactly, with no gaps or post-hoc adjustments; without this verification the combinatorial proof is incomplete.

Simulated Author's Rebuttal

1 responses · 0 unresolved

Thank you for the referee's thoughtful comments on our manuscript. We appreciate the emphasis on ensuring the combinatorial proof is fully explicit. We respond to the major comment as follows.

read point-by-point responses

  1. Referee: [Abstract, paragraph 3 (and the decomposition section that follows)] The central claim rests on the assertion (abstract, paragraph 3) that the recursive decomposition defined for greedy Tamari intervals is isomorphic to the standard decomposition of bipartite planar maps, including preservation of the refined enumeration parameters. The manuscript must supply an explicit, case-by-case verification that every interval decomposes uniquely into a root component plus a sequence of smaller intervals whose sizes and labels match the map decomposition exactly, with no gaps or post-hoc adjustments; without this verification the combinatorial proof is incomplete.

    Authors: We agree that an explicit verification strengthens the presentation. In the decomposition section, we define the recursive structure by identifying the root edge and the attached subintervals, with parameters tracking the sizes and the refined statistics (such as the number of certain vertices or edges). The uniqueness follows from the greedy property of the Tamari intervals, which determines the decomposition uniquely. The isomorphism is established by showing that this decomposition mirrors exactly the standard one for bipartite planar maps, where the root component corresponds to the root face or edge, and submaps to subintervals. To make this more transparent, we will revise the manuscript to include a proposition that lists the cases (base case of single interval, and recursive cases based on the structure) and verifies the matching of sizes, labels, and statistics in each case. This will confirm there are no gaps or adjustments needed. revision: yes

Circularity Check

0 steps flagged

No circularity; independent combinatorial decomposition and isomorphism.

full rationale

The paper establishes an explicit recursive decomposition for greedy Tamari intervals and proves it isomorphic to the standard decomposition of bipartite planar maps, yielding the refined enumeration. This construction is presented as a direct combinatorial argument, not derived from fitted parameters, self-citations, or prior ansatzes by the same authors. No equations or steps reduce by construction to the target count; the isomorphism is the independent content of the proof. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no free parameters, axioms, or invented entities are identifiable from the provided text.

pith-pipeline@v0.9.1-grok · 5669 in / 1061 out tokens · 29111 ms · 2026-07-02T09:54:26.757172+00:00 · methodology

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Reference graph

Works this paper leans on

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