A generalised Euler-Poincar\'e formula for associahedra

TL;DR

This paper generalizes the Euler-Poincaré formula to count flip-equivalence classes of polygon tilings based on integer partitions, extending the classical associahedra case.

Contribution

It introduces a new generalized Euler-Poincaré formula applicable to associahedra and related tiling configurations.

Findings

Derived a formula for flip-equivalence classes of polygon tilings

Generalized the Euler-Poincaré formula for associahedra

Provides a combinatorial enumeration method

Abstract

We derive a formula for the number of flip-equivalence classes of tilings of an $n$-gon by collections of tiles of shape dictated by an integer partition $\lambda$. The proof uses the Euler-Poincar\'e formula; and the formula itself generalises the Euler-Poincar\'e formula for associahedra.