Proofs of four generating function conjectures for arbor polytopes

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.

• The paper establishes closed-form generating functions for four key invariants (Zeta, M-triangle, Ehrhart, and Laplace-transformed volume) of arbor polytopes.
• It employs combinatorial techniques, operator calculus, and recursive methods to derive precise enumerative formulas.
• The results provide a foundational framework connecting lattice polytope geometry and poset invariants with implications for further algebraic and topological research.

Proofs of Generating Function Conjectures for Arbor Polytopes

Introduction

The paper "Proofs of four generating function conjectures for arbor polytopes" (2605.08968) rigorously establishes four conjectures formulated by Chapoton, which concern fundamental generating series associated with the lattice structure and geometry of a specific sequence of arbor polytopes. Arbor polytopes, rooted in the combinatorial framework connecting rooted trees (arbors) to lattice polytopes and graded posets, encode nontrivial enumerative and geometric data. The proven conjectures pertain to the generating series of the Zeta polynomial and M-triangle for associated posets, and the Ehrhart polynomial and Laplace-transformed volume for the polytope sequence defined by elementary arbors.

Arbor Polytope Structure and Poset Invariants

The sequence of arbors considered are trees with a root labeled by a singleton subset of $[n]$ and $n-1$ child vertices, each being a singleton leaf. The associated lattice polytope $Q_t$ is defined in $\mathbb{R}^n$ by the inequalities $x_i \ge 0$ for $i \in [n]$ and $x_{i} \le \sum_{j \in D(v)} x_j$ per vertex $v$, where $D(v)$ is the label set of the subtree rooted at $v$. The induced poset $n-1$0 consists of the lattice points in $n-1$1 ordered componentwise. These structures are notable for their simplicity yet nontrivial enumerative properties, serving as a bridge between combinatorial, geometric, and algebraic enumerative frameworks.

The paper focuses on four key invariants:

• The Zeta polynomial $n-1$2 counts the number of chains of length $n-1$3 in $n-1$4.
• The M-triangle $n-1$5 is a generating function involving Möbius numbers and the rank function of the poset.
• The Ehrhart polynomial $n-1$6 counts lattice points in the $n-1$7-th dilation $n-1$8.
• The Laplace transform of the volume function $n-1$9 refines the volume distribution over height slices.

Proof of the Zeta Polynomial Generating Function

The first main result provides a closed-form expression for the generating series

$Q_t$0

This formula arises via an explicit summation over chain heights and uses elementary combinatorial identities. The derivation leverages recurrence relations on the poset structure, combined with generating function manipulations, including transformation and summation over possible chain contributions.

Explicit Generating Function for the M-Triangle

The generating series for the M-triangle associated with the poset $Q_t$1 is established as

$Q_t$2

This result is obtained by exploiting an explicit formula for the generalized poset weight enumerator $Q_t$3 and the known transformation between $Q_t$4 and $Q_t$5, specifically $Q_t$6. The algebraic complexity is handled by formal manipulations of bivariate generating functions and recursive combinatorial arguments.

Ehrhart Polynomial Generating Function

The generating function for the Ehrhart polynomials is provided in closed-form as

$Q_t$7

or, equivalently, in the compact rational form detailed in the text. The paper presents two independent proofs: the first via weighted lattice-point enumeration polynomials (exploiting the structure of the arbor as a product of intervals) and generating function calculus, and the second by inclusion-exclusion over the facets, referencing prior explicit results [see also (Athanasiadis et al., 12 Mar 2026)]. Both approaches exploit a unified multiplicative structure intrinsic to this class of arbors.

Laplace Transform of the Volume Generating Series

For the Laplace transform of the volume of slices, the generating function is proved as

$Q_t$8

Here, $Q_t$9 and $\mathbb{R}^n$0 encode the Laplace variable substitution. The proof employs a recursive operator construction, leveraging a combinatorial truncation operator $\mathbb{R}^n$1 (acting on polynomial powers) that respects the decomposition and supports a recurrence on the size of the arbor.

Theoretical and Practical Implications

The rigorous confirmation of Chapoton's four conjectures provides a comprehensive framework for tractable families of lattice polytopes derived from tree combinatorics. The explicit generating functions enable closed-form enumeration of chains, volumes, and lattice points for this class, facilitating further analysis in algebraic combinatorics, polyhedral geometry, and their connections to algebraic topology (notably in Hochschild and loop space models).

The utilization of operator calculus and generating function techniques yields methodologies adaptable to broader classes of polytopes and posets. Explicit volume formulas and their transforms are crucial for precise enumerative and asymptotic analysis. The results lay groundwork for algorithmic enumeration via symbolic computation systems (e.g., FriCAS), as referenced.

Future Directions

Potential extensions include generalization to more complex arbor structures, exploration of other weighted and multivariate invariants, and the study of relations with permutation and partition polytopes. Connections to representation theory (via poset and polytope invariants) merit systematic investigation, particularly in the context of categorical and topological analogs. Algorithmic enhancement for lattice point enumeration in similarly structured polytopes remains an open computational avenue.

Conclusion

This paper closes a set of open conjectures regarding generating functions of poset and polytope invariants for elementary arbor polytopes, offering explicit closed formulas and characterizing the algebraic essence of the enumerative combinatorics inherent to these structures. The results consolidate the theoretical foundation for deeper study of the interplay between poset invariants, polytopal geometry, and combinatorial generating series.