The authors give a very natural description of the bijections between the set of cells in the minimal CW-complex homotopy equivalent to the complement of a complexified real supersolvable arrangement $\mathcal{A}$, the \textbf{nbc}-basis (non broken circuit basis) of the Orlik-Solomon algebra associated to $\mathcal{A}$ and the set of chambers of $\mathcal{A}$. They use these bijections to describe a bijection between the symmetric group and the \textbf{nbc}-basis of the braid arrangement.