The coarse flag Hilbert-Poincaré series of the braid arrangement

The paper concerns the coarse flag Hilbert-Poincaré series of Maglione-Voll in the case of the braid arrangement associated to the symmetric group. We explicitly construct a companion statistic $\mathrm{ino}:{\mathfrak{S}}_{n+1}\times \mathrm{Sym}(n)\to \mathbb{N}$ for the descent statistic on $\mathrm{Sym}(n)$ using reverse $(P,\omega )$-partitions and quasisymmetric functions.