Chains in shard lattices and BHZ posets

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.