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arxiv: 2605.29431 · v1 · pith:M2RNGD4Bnew · submitted 2026-05-28 · 🧮 math.CO

Rowmotion on hook and two-row alt ν-Tamari lattices

Sen-Peng Eu , Vei-Cheng Hioe , Yi-Lin Lee This is my paper

Pith reviewed 2026-06-29 06:51 UTC · model grok-4.3

classification 🧮 math.CO
keywords rowmotionalt ν-Tamari latticeshook-Tamari latticestwo-row Tamari latticescyclic sievingorbit structureshomometric statisticssemidistributive lattices
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The pith

Orbit structures under rowmotion on alt hook-Tamari and two-row Tamari lattices stay the same regardless of the increment vector δ.

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

The paper studies rowmotion, the operation that sends each element of a poset to the unique minimal element above it in a certain covering relation, on two families of alt ν-Tamari lattices. It determines the cycle decompositions explicitly for the hook case (ν = EN^{a-1}E^{b-1}N) and the two-row case (ν = E^a N E^b N) and proves these decompositions do not change when δ is varied. The independence immediately yields that rowmotion on the hook family satisfies the cyclic sieving phenomenon. The authors further compute the sums of several statistics over each orbit and establish that down-degree, peak, and valley sums are constant across orbits while the area sum is not.

Core claim

We explicitly determine the orbit structures of H_δ(a,b) and T_δ(a,b) under rowmotion, and prove that their orbit structures are independent of the increment vector δ. As a consequence, we show that rowmotion on H_δ(a,b) exhibits the cyclic sieving phenomenon. We also compute orbit sums for several natural statistics; all of these except the area statistic are homometric under rowmotion.

What carries the argument

The switching property for semidistributive lattices, which equates the orbit structures arising from different increment vectors by local comparison of covering relations.

If this is right

  • Rowmotion on every alt hook-Tamari lattice H_δ(a,b) satisfies cyclic sieving for any choice of δ.
  • The sums of the down-degree, peak, and valley statistics over orbits are independent of δ in both families.
  • Orbit sums can be computed once for a convenient increment vector and then reused for all others.
  • The homometric property holds for the listed statistics on the hook and two-row families.

Where Pith is reading between the lines

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

  • The same independence argument may apply to other families of alt ν-Tamari lattices once a suitable switching relation is identified.
  • Homometric statistics under rowmotion could be used to produce new q-analogues or generating functions that are invariant under the action.
  • The local modification technique used for the hook Hasse diagrams might extend to other graded posets with simple covering relations.

Load-bearing premise

The switching property introduced for semidistributive lattices holds and lets one transfer orbit information directly between different increment vectors in the two-row setting.

What would settle it

An explicit pair of distinct increment vectors δ and δ' together with parameters a,b such that the multiset of orbit lengths under rowmotion on H_δ(a,b) differs from the multiset on H_δ'(a,b).

Figures

Figures reproduced from arXiv: 2605.29431 by Sen-Peng Eu, Vei-Cheng Hioe, Yi-Lin Lee.

Figure 1
Figure 1. Figure 1: (a) (resp., [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The Hasse diagram of the alt hook-Tamari lattice Tamδ(ν), where ν = EN2E 2N and (a) δ = (0, 0), (b) δ = (1, 0), and (c) δ = (2, 0). The coordinate of each element is also given. Next, given a ν-path µ, the ν-bracket vector of µ is a vector b(µ) = (b0(µ), . . . , br(µ)) obtained from the following procedures: (1) Set bfi (µ) = i for each i = 0, 1, . . . , n. (2) Let gi be the number of i’s in the vector a(µ… view at source ↗
Figure 3
Figure 3. Figure 3: A semidistributive lattice L with 13 elements. 2.4 The cyclic sieving phenomenon In their seminal paper, Reiner, Stanton, and White [13] introduced the cyclic sieving phenomenon (CSP), which provides a unified way to describe the orbit structure of a set of combinatorial objects under a cyclic action. Formally, let X be a finite set, and g ∶ X → X be an invertible map of order n, that is, n is the smallest… view at source ↗
Figure 4
Figure 4. Figure 4: An element µ (drawn in red) of the alt hook-Tamari lattice Hδ(k)(a, b) with (a, b, k) = (4, 7, 3). The element µ can be identified as an element of the Tamari lattice Tam(̂ν(3)), where the lattice path ̂ν(3) is drawn in blue. Lemma 3.2. Let a, b be positive integers and k be an integer with 0 ≤ k ≤ b − 1. Fix the lattice path ν = ENa−1E b−1N. Let µ be an element of Hδ(k)(a, b). Then the ̂ν(k)-bracket vecto… view at source ↗
Figure 5
Figure 5. Figure 5: The Hasse diagram of the alt hook-Tamari lattice Hδ(k)(a, b) with (a, b) = (8, 6) and (a) k = 0, (b) k = 2. When a = 1, it follows again from Lemma 3.2 that the simplified ̂ν(k)-bracket vectors correspond to the lattice points {(0, t)∣0 ≤ t ≤ b − 1} ∪ {(1, b − 1)}. Lemma 3.3 implies that the lattice structure of Hδ(1, b) is the chain of b + 1 elements, which agrees with our statement (the k = 0 case). 3.2 … view at source ↗
Figure 6
Figure 6. Figure 6: An element µ (drawn in red) of the alt 2-row-Tamari lattice Tδ(k)(a, b) with (a, b, k) = (3, 3, 2). The element µ can be identified as an element of the Tamari lattice Tam(̂ν(2)), where the lattice path ̂ν(2) is drawn in blue. Lemma 4.2. Let a, b and k be nonnegative integers with 0 ≤ k ≤ b. Fix the lattice path ν = E aNEbN. Let µ be an element of Tδ(k)(a, b). Then the ν̂(k)-bracket vector of µ has the fol… view at source ↗
Figure 7
Figure 7. Figure 7: The Hasse diagram of the alt 2-row-Tamari lattice Tδ(k)(a, b) with (a, b) = (5, 3) and (a) k = 0, (b) k = 2. Proof. By Lemma 4.2 and the discussion before Lemma 4.3, we may express each element µ of Tδ(k)(a, b) as a triple (a−s, u, v), where s is the number of zeroes at the non-fixed positions of b(µ) and u (resp., v) is the number of 2’s in α (resp., β). Recall that the possible pairs (u, v) are given in … view at source ↗
Figure 8
Figure 8. Figure 8: An illustration of forming pentagons in the Hasse diagram of Tδ(k)(a, b). 4.2 The invariance of rowmotion on Tδ(a, b) In this section, we prove that the orbit structure of the alt 2-row-Tamari lattice Tδ(k)(a, b) under rowmotion is independent of δ(k). The statistic ddeg over each such orbit is independent of δ(k) as well. We begin with the following definition. Definition 4.5. Let (L, ≤) be a semidistribu… view at source ↗
read the original abstract

In 2024, Ceballos and Chenevi{\`e}re introduced alt $\nu$-Tamari lattices, parameterized by a lattice path $\nu$ and an increment vector $\delta$, as a common generalization of $\nu$-Tamari and $\nu$-Dyck lattices. We study rowmotion on two families: the alt hook-Tamari lattice $\mathsf{H}_{\delta}(a,b)$ (where $\nu=EN^{a-1}E^{b-1}N$) and the alt $2$-row-Tamari lattice $\mathsf{T}_{\delta}(a,b)$ (where $\nu=E^aNE^bN$). We explicitly determine the orbit structures of $\mathsf{H}_{\delta}(a,b)$ and $\mathsf{T}_{\delta}(a,b)$ under rowmotion, and prove that their orbit structures are independent of the increment vector $\delta$. As a consequence, we show that rowmotion on $\mathsf{H}_{\delta}(a,b)$ exhibits the cyclic sieving phenomenon. We also compute orbit sums for several natural statistics. In the hook case, we evaluate the down-degree, peak, valley, and area statistics; in the $2$-row case, we focus on the down-degree statistic. All of these -- except for the area statistic -- are homometric under rowmotion. Regarding the methodology of this paper, our results in the hook case are obtained by applying a simple local modification to their Hasse diagrams. In the $2$-row case, we introduce a switching property for semidistributive lattices, which allows us to compare the orbit structures arising from different increment vectors.

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 manuscript determines the orbit structures of rowmotion on the alt hook-Tamari lattices H_δ(a,b) (ν = EN^{a-1}E^{b-1}N) and alt two-row Tamari lattices T_δ(a,b) (ν = E^a N E^b N). It proves these structures are independent of the increment vector δ, establishes that rowmotion on H_δ(a,b) exhibits the cyclic sieving phenomenon, and computes orbit sums for the down-degree, peak, valley, and area statistics in the hook case and down-degree in the two-row case, showing homometry under rowmotion for all except area. Proofs rely on local Hasse-diagram modifications for the hook family and a newly introduced switching property for semidistributive lattices in the two-row case.

Significance. If the claims hold, the work supplies explicit orbit data for rowmotion on these generalized Tamari lattices and introduces a switching property that enables parameter-independent comparisons. The δ-independence result and the CSP corollary are substantive if the arguments are complete; the homometric statistics provide additional concrete information on statistic behavior. The local-modification approach for hooks is self-contained and the switching property may have wider use in lattice dynamics.

major comments (1)
  1. [switching property definition and application (two-row case)] The switching property for semidistributive lattices (introduced to equate orbit structures across different δ in the two-row case) is load-bearing for the δ-independence claim on T_δ(a,b) and the subsequent CSP corollary. The manuscript must verify explicitly that the property transfers the full cycle index (cycle lengths and multiplicities), not merely a coarser invariant such as the number of orbits.
minor comments (2)
  1. Notation for the lattices is introduced as alt ν-Tamari in the abstract but the title uses alt ν-Tamari; ensure consistent spelling and capitalization throughout.
  2. The abstract states that all statistics except area are homometric; the manuscript should clarify whether this holds uniformly for both families or only where computed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for highlighting the need for explicit verification of the switching property. We address the single major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: The switching property for semidistributive lattices (introduced to equate orbit structures across different δ in the two-row case) is load-bearing for the δ-independence claim on T_δ(a,b) and the subsequent CSP corollary. The manuscript must verify explicitly that the property transfers the full cycle index (cycle lengths and multiplicities), not merely a coarser invariant such as the number of orbits.

    Authors: The switching property (Definition 4.2) is constructed as a bijection φ between the underlying sets of T_δ(a,b) and T_δ'(a,b) that is a lattice isomorphism for the semidistributive structure and satisfies φ ∘ rowmotion_δ = rowmotion_δ' ∘ φ. Because rowmotion is defined uniformly via the join and meet operations of the lattice, the commuting relation directly implies that φ maps orbits to orbits while preserving their lengths and multiplicities; the cycle index is therefore identical. The explicit orbit structures are first computed for a convenient choice of δ and then transferred via this isomorphism. We agree that a dedicated lemma stating “the switching map induces an equivariant bijection of dynamical systems, hence identical cycle indices” would make the argument fully transparent, and we will insert this lemma (with a short proof) in the revised version. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation self-contained via new combinatorial arguments

full rationale

The paper introduces and applies a switching property on semidistributive lattices to equate orbit structures across increment vectors δ in the two-row case T_δ(a,b), while the hook case H_δ(a,b) uses independent local Hasse-diagram modifications. These steps are presented as newly proven combinatorial facts rather than reductions to prior fits, self-definitions, or self-citation chains; the δ-independence and cyclic sieving claims therefore rest on self-contained arguments without the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

No free parameters or invented entities appear in the abstract. The work relies on standard properties of semidistributive lattices and poset operations.

axioms (1)
  • domain assumption Semidistributive lattices admit a switching property that preserves orbit structures under rowmotion for different increment vectors
    Invoked to compare orbits across δ values in the two-row case.

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Reference graph

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