# On Andrews' integer partitions with even parts below odd parts

## TL;DR

This paper investigates the properties of Andrews' partition functions related to partitions with specific even and odd part constraints, establishing congruences, parity results, and divisibility properties using modular form techniques.

## Contribution

It proves infinite families of congruences and parity results for the partition function ar{ ext{EO}}(n), and demonstrates divisibility properties of Uncu's related partition function.

## Key findings

- Infinite congruences for ar{ ext{EO}}(n)
- Infinitely many even and odd values of ar{ ext{EO}}(n) in arithmetic progressions
- Divisibility by powers of 2 for Uncu's partition function

## Abstract

Recently, Andrews defined a partition function $\mathcal{EO}(n)$ which counts the number of partitions of $n$ in which every even part is less than each odd part. He also defined a partition function $\overline{\mathcal{EO}}(n)$ which counts the number of partitions of $n$ enumerated by $\mathcal{EO}(n)$ in which only the largest even part appears an odd number of times. Andrews proposed to undertake a more extensive investigation of the properties of $\overline{\mathcal{EO}}(n)$. In this article, we prove infinite families of congruences for $\overline{\mathcal{EO}}(n)$. We next study parity properties of $\overline{\mathcal{EO}}(n)$. We prove that there are infinitely many integers $N$ in every arithmetic progression for which $\overline{\mathcal{EO}}(N)$ is even; and that there are infinitely many integers $M$ in every arithmetic progression for which $\overline{\mathcal{EO}}(M)$ is odd so long as there is at least one. Very recently, Uncu has treated a different subset of the partitions enumerated by $\mathcal{EO}(n)$. We prove that Uncu's partition function is divisible by $2^k$ for almost all $k$. We use arithmetic properties of modular forms and Hecke eigenforms to prove our results.