On Stanley's partition function (Q1960265)

Summary: \textit{R.P. Stanley} [``Problem 10969,'' Am. Math. Monthly 109, 760 (2002); ``Some remarks on sign-balanced and maj-balanced posets,'' Adv. Appl. Math. 34, No.\, 4, 880--902 (2005; Zbl 1097.06004)] defined a partition function \(t(n)\) as the number of partitions \(\lambda\) of \(n\) such that the number of odd parts of \(\lambda\) is congruent to the number of odd parts of the conjugate partition \(\lambda'\) 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 \(p(n)\). A simple combinatorial explanation of this fact is also provided.