  We give a bijection between certain colored partitions and the elements in
the quotient of an affine Weyl group modulo its Weyl group. By Bott's formula
these colored partitions give rise to some partition identities. In certain
types, these identities have previously appeared in the work of
Bousquet-Melou-Eriksson, Eriksson-Eriksson and Reiner. In other types the
identities appear to be new. For type $A_{n}$, the affine colored partitions
form another family of combinatorial objects in bijection with $n+1$-core
partitions and $n$-bounded partitions. Our main application is to characterize
the rationally smooth Schubert varieties in the affine Grassmannians in terms
of affine partitions and a generalization of Young's lattice which refines weak
order and is a subposet of Bruhat order. Several of the proofs are computer
assisted.
