F-triangle realization for Hochschild polytopes

Construct, for every dimension n, a maximal face C of the polar simplicial complex SC_n of the Hochschild polytope such that the F-triangle of the pair (SC_n,C) coincides with the F-triangle obtained from the transmuted M-triangle of the arbor poset P_{t_n}.

Background

The paper considers arbors t_n consisting of a root vertex with one element and n−1 attached vertices, and relates their posets P_{t_n} to Hochschild polytopes. After proving equality of the numbers of vertices and equality of h-vectors, it conjectures a finer relation between the polar dual of the Hochschild polytope and the transmuted M-triangle of P_{t_n}. The proposed relation has been checked computationally for n at most 9.

References

In the simplicial complex $SC_{n}$, there exists a maximal face $C$ such that the $F$-triangle for the pair $(SC_{n}, C)$ coincides with the $F$-triangle associated with the transmuted $M$-triangle of $P_{t_n}$.

— On posets and polytopes attached to arbors  (2503.04247 - Chapoton, 6 Mar 2025) in Conjecture, Section 9, “Hochschild polytopes”