  For every finite Coxeter group W , we prove that the number of chains in the
shard intersection lattice introduced by Reading on the one hand and in the BHZ
poset introduced by Bergeron, Zabrocki and the third author on the other hand,
are the same. We also show that these two partial orders are related by an
equality between generating series for their M{\"o}bius numbers, and provide a
dimension-preserving bijection between the order complex on the BHZ poset and
the pulling triangulation of the permutahedron arising from the right weak
order, analogous to the bijection defined by Reading between the order complex
of the shard poset and the same triangulation of the permutahedron.
