  We investigate the connections between flavored quivers, dimer models, and
BPS pyramids for generic toric Calabi-Yau threefolds from various perspectives.
We introduce a purely field theoretic definition of both finite and infinite
pyramids in terms of quivers with flavors. These pyramids are associated to the
counting of BPS invariants for generic toric Calabi-Yau threefolds. We discuss
how cluster transformations provide an efficient recursive method for computing
pyramid partition functions and show that the recursion is equivalent to the
multidimensional octahedron recurrence. Transitions between different pyramids
are related to Seiberg dualities, and we offer complimentary characterizations
of these transitions in terms of the motion of zonotopes and duality webs. Our
methods apply to completely general geometries including those with vanishing
4-cycles, which are associated to chiral quivers, thus overcoming one of the
main limitations in the existing literature. We illustrate our ideas with
explicit results for the infinite family of L^{a,b,c} geometries, dP_2,
pseudo-dP_2, and dP_3. The counting of pyramid partitions for dP_1 gives rise
to the Somos-4 sequence, while dP_2 and pseudo-dP_2 generate the Somos-5
sequence. Our results for dP_3 reproduce and extend those previously obtained
for this theory, which were originally obtained from dimer shuffling.
