  A basis of the center of the 0-Hecke algebra of an arbitrary finite Coxeter
group was described by He in 2015. This basis is indexed by certain equivalence
classes of the Coxeter group whose explicit description is rather complicated.
Even their number is not obvious.
  We consider case of the symmetric group $\mathfrak{S}_n$. Building on work of
Geck, Kim and Pfeiffer, we obtain a complete set of representatives of the
equivalence classes. This set is naturally parametrized by certain compositions
of $n$ called maximal. It follows that the maximal compositions of $n$ index
the basis of the center of the 0-Hecke algebra of $\mathfrak{S}_n$. We then
develop an explicit combinatorial description for the equivalence classes that
are parametrized by the maximal compositions whose odd parts form a hook.
