Generating series for Zeta polynomials of Hochschild arbor posets

Derive and prove the conjectural generating-series identity for the refined Zeta polynomials of the Hochschild arbor posets P_{t_n}, namely the identity displayed in the paper for 1+Σ_{n≥1} 𝖹_{t_n}(u,1)s^n.

Background

The paper introduces the arbor family t_n associated with Hochschild polytopes and computes several low-dimensional examples. Using computer algebra, the authors formulate a conjectural closed generating series for the refined Zeta polynomials of P_{t_n}. The identity is not derived in the paper and is presented as a conjectural formula.

References

Using \verb|FriCAS| , one can guess the following conjectural formulas for various generating series in $s$ for the arbors $t_n$. For the Zeta polynomials of the posets $P_{t_n}$:

1 + \sum_{n \geq 1} \mathsf{Z}_{t_n}(u, 1) sn \stackrel{?}{=}\n\exp\left( \int \frac{u}{(1-us)(1+s-us)} ds \right).\n

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