Partitions which are simultaneously \(t_1\)- and \(t_2\)-core
For any node in the Ferrers-Young diagram of a partition of an integer \(n\), define the hook number of the node to be \(1+ r+ d\) where \(r\), \(d\) are the numbers of nodes to its right, downwards respectively. A partition is \(t\)-core if none of its hook numbers is divisible by \(t\). It is shown that if \(t_1\), \(t_2\) are relatively prime then there are precisely \({1\over t_1+ t_2} {t_1+ t_2\choose t_1}\) partitions which are both \(t_1\)-core and \(t_2\)-core. If \(t_1\), \(t_2\) are not relatively prime, there are infinitely many such partitions.
scientific article; zbMATH DE number 1746262