  We give relations between the joint distributions of multiple hook lengths
and of frequencies and part sizes in partitions, extending prior work in this
area. These results are discovered by investigating truncations of the
Han/Nekrasov-Okounkov hooklength formula and of (k,j)-colored partitions, a
unification of k-colored partitions and overpartitions. We establish the
observed relations at the constant and linear terms for all n, and for j=2 in
their quadratic term, with the associated hook/frequency identities. Further
results of this type seem likely.
