Generating series for M-triangles of Hochschild arbor posets

Derive and prove the conjectural generating-series identity for the M-triangles of the Hochschild arbor posets P_{t_n}, namely the identity displayed in the paper for 1+Σ_{n≥1}M_{t_n}(X,Y)s^n.

Background

For the Hochschild arbor family t_n, the paper develops recursive formulas for M-triangles and verifies the proposed relationship with Hochschild-lattice data in small dimensions. Computer experimentation suggests a closed generating series for the entire family of M-triangles. The displayed identity is explicitly marked conjectural and remains unproved.

References

For the $M$-triangles of the posets $P_{t_n}$:

1 + \sum_{n \geq 1} M_{t_n}(X,Y) sn \stackrel{?}{=} \frac{\left(X Y s - Y s - 1\right)} {\left(X Y s - 1\right)}{\left(2 \, X Y s - Y s - 1\right)} {\left(X Y s - Y s + s - 1\right)}.\n

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