  It is well known that in string compactifications on toric Calabi-Yau
manifolds one can introduce refined BPS invariants that carry information not
only about the charge of the BPS state but also about the spin content. In this
paper we study how these invariants behave under wall crossing. In particular,
by applying a refined wall crossing formula, we obtain the refined BPS
degeneracies for the conifold in different chambers. The result can be
interpreted in terms of a new statistical model that counts `refined' pyramid
partitions; the model provides a combinatorial realization of wall crossing and
clarifies the relation between refined pyramid partitions and the refined
topological vertex. We also compare the wall crossing behavior of the refined
BPS invariants with that of the motivic Donaldson-Thomas invariants introduced
by Kontsevich-Soibelman. In particular, we argue that, in the context of BPS
state counting, the three adjectives in the title of this paper are essentially
synonymous.
