Trimming the permutahedron to extend the parking space

Berget and Rhoades asked whether the permutation representation obtained by action of $S_{n-1}$ on parking functions length $n-1$ can be extended to a $S_{n}$. We answer this question in affirmative. realize our module two different ways. The first description involves binary Lyndon words second symmetric group lattice points trimmed standard permutahedron.