Tuesday May 30

The hive -> honeycomb bijection.
The puzzle -> honeycomb injection, and its image.

c_lambda,mu^nu nonzero
<=> there exist integral hives with that boundary
=====> there exist real hives with that boundary
<=> that boundary satisfies all puzzle inequalities
<=> for certain smaller integer honeycombs, the boundary satisfies certain inequalities
<=> for certain smaller (lambda',mu',nu') with c_lambda',mu'^nu' nonzero, (lambda,mu,nu) satisfy certain inequalities.

This becomes a recurrence once one can reverse the long arrow, which is now called the saturation theorem. The recursion was conjectured by Horn, in the context of the sums of Hermitian matrices.