arxiv: 2605.08968 · v1 · submitted 2026-05-09 · 🧮 math.CO

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· Lean Theorem

Proofs of four generating function conjectures for arbor polytopes

Feihu Liu, Jinlong Tang

Pith reviewed 2026-05-12 02:21 UTC · model grok-4.3

classification 🧮 math.CO

keywords arbor polytopesgenerating functionszeta polynomialsM-trianglesEhrhart polynomialsposetscombinatorial invariants

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The pith

Four conjectured generating series for arbor posets and polytopes are proven to have explicit closed forms.

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

The paper proves four generating-function conjectures originally posed by Chapoton for a sequence of arbors. Two of the results give closed forms for the generating series of the zeta polynomial and of the M-triangle of the associated poset. The other two give closed forms for the Ehrhart polynomial and for the Laplace transform of the volume function of the associated arbor polytope. These formulas supply exact expressions for counting and volume invariants that previously lacked simple descriptions.

Core claim

The generating series of the zeta polynomial of the arbor poset, the generating series of its M-triangle, the Ehrhart polynomial of the arbor polytope, and the Laplace transform of the volume function of that polytope each admit the specific closed-form expressions conjectured by Chapoton. The proofs establish these identities by working directly with the combinatorial structure of the given sequence of arbors.

What carries the argument

The sequence of arbors, which simultaneously defines a graded poset (whose chain and order-ideal data yield the zeta polynomial and M-triangle) and a lattice polytope (whose lattice-point enumerator and volume yield the Ehrhart polynomial and Laplace transform).

Load-bearing premise

The arbor posets and polytopes are defined exactly as in Chapoton's original conjectures, and the combinatorial identities used in the proofs hold without additional unstated restrictions on the sequence of arbors.

What would settle it

Direct computation of the zeta polynomial (or M-triangle) for the first few arbors in the sequence, followed by coefficient comparison against the conjectured closed-form series.

Figures

Figures reproduced from arXiv: 2605.08968 by Feihu Liu, Jinlong Tang.

**Figure 1.** Figure 1: An arbor on the set [8]. Definition 1.1 (The Polytope Qt). The polytope Qt is defined in the vector space R n with coordinates (xi)i∈[n] by the inequalities xi ≥ 0 for all i ∈ [n] and X i∈D(v) xi ≤ |D(v)| for every vertex v, where D(v) is the set of labels in the sub-tree rooted at v. For the example in [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** Figure 2: The example of t5. The arbor polytope Qtn is also a special case of a class of polytopes studied in the literature [2]. Using FriCAS [7], Chapoton [4, Section 9] conjectured four formulas for various generating series in s for the arbors tn. The four formulas respectively involve • the Zeta polynomials of the posets Ptn (see Theorem 2.2), • the M-triangles of the posets Ptn (see Theorem 2.6), • the Ehrhart… view at source ↗

read the original abstract

This paper proves four conjectured generating series, due to Chapoton, which concern invariants of posets and polytopes associated with a specific sequence of arbors. Two of these conjectures provide closed-form formulas for the generating series of the Zeta polynomial and the generating series of the M-triangle of the poset, respectively. The remaining two conjectures pertain, respectively, to the Ehrhart polynomial and the Laplace transform of the volume function of the associated arbor polytope.

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

This paper proves four open conjectures by Chapoton, giving closed forms for the zeta polynomial, M-triangle, Ehrhart polynomial, and Laplace transform of volume for arbor posets and polytopes.

read the letter

This paper proves four conjectures by Chapoton on generating functions tied to arbor posets and their polytopes. It delivers explicit formulas for the generating series of the zeta polynomial and the M-triangle of the poset, along with the Ehrhart polynomial and the Laplace transform of the volume function for the polytope. Those closed forms were the open parts, and the work supplies them directly from the definitions in the earlier conjectures.

Referee Report

0 major / 2 minor

Summary. The manuscript proves four conjectures due to Chapoton by establishing closed-form generating series for the Zeta polynomial and M-triangle of the arbor poset, together with the Ehrhart polynomial and the Laplace transform of the volume function of the associated arbor polytope.

Significance. Resolving these conjectures supplies explicit formulas for key invariants of arbor posets and polytopes, enabling direct computation and further theoretical work in combinatorial enumeration and Ehrhart theory. The combinatorial proofs, once verified, constitute a concrete advance by converting open questions into established results.

minor comments (2)

The introduction would benefit from a short paragraph recalling the precise definition of the sequence of arbors used in Chapoton's original conjectures, to make the matching explicit.
Ensure that all generating-function identities are accompanied by a clear statement of the range of summation or the initial conditions employed.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending minor revision. No specific major comments were provided in the report, so there are no individual points requiring a point-by-point response. We will incorporate any minor editorial or presentational suggestions in the revised version.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper proves four external conjectures posed by Chapoton on generating functions for the Zeta polynomial, M-triangle, Ehrhart polynomial, and Laplace transform of the volume for arbor posets and polytopes. The abstract and summary indicate the results are established via combinatorial proofs of these independent conjectures rather than any reduction of outputs to fitted inputs, self-definitions, or self-citation chains within the paper. No load-bearing steps reduce by construction to the paper's own quantities, so the derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, no explicit free parameters, ad-hoc axioms, or invented entities are described; the work relies on standard combinatorial definitions of posets, polytopes, and generating functions from prior literature.

pith-pipeline@v0.9.0 · 5360 in / 996 out tokens · 43847 ms · 2026-05-12T02:21:59.659366+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
The arbor polytope Q_t is defined... inequalities x_i >=0 and sum_{i in D(v)} x_i <= |D(v)| ... poset P_t ordered by coordinate-wise comparison... Zeta polynomial Z_t(u,X), M-triangle M_t(X,Y), Ehrhart E_t(u), Laplace L_t(v)
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
Z_tn(u,1) = sum binom(n-1,k) (u-1)^k binom(u+n-k-1,n-k) ... generating series exp(integral) = ((1-su+s)/(1-su))^u

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

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