On Stanley's Partition Function

TL;DR

This paper investigates a special partition function t(n) related to odd parts and hooks, providing a closed-form generating function and revealing its parity relation to the ordinary partition function p(n).

Contribution

It introduces a new partition function t(n), establishes its equivalence to partitions with even hooks of even length, and derives its generating function.

Findings

t(n) equals the number of partitions with an even number of hooks of even length

t(n) has the same parity as p(n) for all n

A combinatorial explanation for the parity relation is provided

Abstract

Stanley 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) for any n. A simple combinatorial explanation of this fact is also provided.