For any $n\in\mathbb{N}$, the permutahedron is the convex polytope (for $S_n$ the symmetric group). $$ P_n = \mathsf{conv}(\sigma((1,2,\dots,n)) \mid \sigma\in S_n\} $$ i.e. it is given by the convex hull of all permutations of some fixed coordinate vector $(1,2,\dots,n)$.
For a uniform random variable $U\sim \mathsf{Unif}(P_n)$, what is known about the 1-dimensional marginals of $P_n$? By this, I mean the distribution induced by projecting $P_n$ onto any particular coordinate axis.