On Stanley's Partition Function

Stanley defined a partition function t(n) as the number of partitions $λ$ of n such that the number of odd parts of $λ$ is congruent to the number of odd parts of the conjugate partition $λ'$ modulo 4. We show that t(n) equals the number of partitions of n with an even number of hooks of even length. We derive a closed-form formula for the generating function for the numbers p(n)-t(n). As a consequence, we see that t(n) has the same parity as the ordinary partition function p(n) for any n. A simple combinatorial explanation of this fact is also provided.

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