AbstractStanley defined a partition function t(n) as the number of partitions λ of n suchthat the number of odd parts of λ is congruent to the number of odd parts of theconjugate partition λ′modulo 4. We show that t(n) equals the number of partitionsof n with an even number of hooks of even length. We derive a closed-form formulafor the generating function for the numbers p(n) − t(n). As a consequence, we seethat t(n) has the same parity as the ordinary partition function p(n). A simplecombinatorial explanation of this fact is also provided...