The centers of gravity of the associahedron and of the permutahedron are the same

TL;DR

This paper proves that Loday's realization of the associahedron shares the same center of gravity as the permutahedron, confirming a conjecture and extending the result to the cyclohedron.

Contribution

It establishes the equality of centers of gravity for specific realizations of associahedron, permutahedron, and cyclohedron, confirming a conjecture and generalizing previous observations.

Findings

Centers of gravity of Loday's associahedron and permutahedron are identical.

The result extends to the cyclohedron with similar realizations.

Confirms a conjecture by F. Chapoton.

Abstract

In this article, we show that Loday's realization of the associahedron has the the same center of gravity than the permutahedron. This proves an observation made by F. Chapoton. We also prove that this result holds for the associahedron and the cyclohedron as realized by the first author and C. Lange.