The poset associahedron $f$-vector is a comparability invariant

TL;DR

This paper proves that the face vector of Galashin's poset associahedron depends solely on the comparability graph of the poset, enabling the construction of polytopes with permutohedron-like face counts that are not combinatorially equivalent.

Contribution

It establishes that the $f$-vector of the poset associahedron is a comparability invariant, revealing new polytopes with permutohedron-like face vectors but different combinatorial structures.

Findings

The $f$-vector depends only on the comparability graph.

Constructs polytopes with permutohedron $f$-vectors that are not combinatorially equivalent.

Shows invariance of the $f$-vector under certain graph transformations.

Abstract

We show that the $f$-vector of Galashin's poset associahedron $\mathscr A(P)$ only depends on the comparability graph of $P$. In particular, this allows us to produce a family of polytopes with the same $f$-vectors as permutohedra, but that are not combinatorially equivalent to permutohedra.