$F$- and $H$-Triangles for $\nu$-Associahedra

TL;DR

This paper introduces two bivariate polynomials, the $F$- and $H$-triangles, associated with $ u$-associahedra, and proves a combinatorial invertible transformation relating them, generalizing classical results in type $A$.

Contribution

It defines new $F$- and $H$-polynomials for $ u$-associahedra and provides a novel combinatorial proof of their invertible relationship, extending classical type $A$ results.

Findings

Established a combinatorial invertible transformation between $F$- and $H$-triangles.

Generalized classical $F$- and $H$-triangles to $ u$-associahedra.

Provided a new combinatorial proof explaining their relationship.

Abstract

For any northeast path $\nu$, we define two bivariate polynomials associated with the $\nu$-associahedron: the $F$- and the $H$-triangle. We prove combinatorially that we can obtain one from the other by an invertible transformation of variables. These polynomials generalize the classical $F$- and $H$-triangles of F.~Chapoton in type $A$. Our proof is completely new and has the advantage of providing a combinatorial explanation of the relation between the $F$- and $H$-triangle.