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For every finite Coxeter group W , we prove that the number of chains in theshard intersection lattice introduced by Reading on the one hand and in the BHZposet 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 anequality between generating series for their M{\"o}bius numbers, and provide adimension-preserving bijection between the order complex on the BHZ poset andthe pulling triangulation of the permutahedron arising from the right weakorder, analogous to the bijection defined by Reading between the order complexof the shard poset and the same triangulation of the permutahedron.

Chains in shard lattices and BHZ posets