Generating series for Ehrhart polynomials of Hochschild arbors

Derive and prove the conjectural generating-series identity for the Ehrhart polynomials of the polytopes Q_{t_n} associated with the Hochschild arbor family t_n, namely the identity displayed in the paper for 1+Σ_{n≥1}Ehr_{t_n}(u)s^n.

Background

The polytopes Q_{t_n} are attached to the arbor family used to model Hochschild polytopes, and the paper provides recursive Ehrhart computations for general arbors. Computational experimentation leads to a proposed generating series for the Ehrhart polynomials of Q_{t_n}. The formula is explicitly marked with a conjectural equality symbol and is not proved.

References

For the Ehrhart polynomials of the polytopes $Q_{t_n}$:

1 + \sum_{n \geq 1} \operatorname{Ehr}_{t_n}(u) sn \stackrel{?}{=}\n\frac{1}{2} \left( 1 -\frac{s}{\left(u s + s - 1\right)} -\frac{s - 1}{\left(u s + s - 1\right)}{2}\right).\n

— On posets and polytopes attached to arbors  (2503.04247 - Chapoton, 6 Mar 2025) in Section 10, final subsection on conjectural formulas