Proofs of four generating function conjectures for arbor polytopes

TL;DR

This paper proves four conjectured generating series related to invariants of posets and arbor polytopes, including formulas for Zeta polynomials, M-triangles, Ehrhart polynomials, and volume transforms.

Contribution

It provides the first proofs of four conjectured generating series for invariants of arbor polytopes and related posets, confirming their closed-form formulas.

Findings

Closed-form formulas for the Zeta polynomial and M-triangle generating series.

Proofs of conjectures related to Ehrhart polynomial and volume transform.

Advancement in understanding invariants of arbor polytopes and associated posets.

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.