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.
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