partitions when $t$ is odd and positive.

    We end by noting $(s,t)$-core and $(\bar{s}, \bar{t})$-core partitions inherit Ramanujan-type congruences from those of $g$-core and $\bar{g}$-core partitions.
address:
- 'Institute of Mathematics, University of Aberdeen, Kings College'
- 'Aberdeen, AB24 3UE, UK'
- 'jbgramain@abdn.ac.uk'
- 'Department of Mathematics, CUNY-York College, Jamaica, NY 11451, USA,'
- 'rnath@york.cuny.edu'
- 'Department of Mathematics, Penn State University, University Park, PA 16802, USA,'
- 'sellersj@psu.edu'
author:
- 'Jean-Baptiste Gramain'
- Rishi Nath
- 'James A. Sellers'
date: '****'
title: Simultaneous core partitions with nontrivial common divisor
---

2010 Mathematics Subject Classification: 05A17, 11P83

Keywords: partition, congruence, generating function

Introduction and background
===========================

partitions, cores and bar-cores
-------------------------------

Let $n$ be a positive integer. A [*partition*]{} $\lambda$ of a nonnegative integer $n$ is a non-decreasing sequence of positive integers, called [*parts*]{}, that sum to $n$. We let $|\lambda|=n$ be the [*size*]{} of $\lambda$. A partition $\lambda$ can be visualized in terms of its [*Young diagram*]{} $[\lambda]$, a graphic representation of $\lambda$ in which left- and top-aligned rows of boxes correspond sequentially to the parts in the partition. \[When the context is clear we will abuse notation and let $\lambda$ refer to both the Young diagram and the 