The $P$-Associahedron $f$-Vector Is a Comparability Invariant

Abstract

For any finite, connected poset $P$, we show that the $f$-vector of Galashin's $P$-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.