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

Colored partitions and the hooklength formula: partition statistic identities