Sách On Stanley’s Partition Function

Abstract
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) ư 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.