Quotient of a partition ($Q_e(\lambda )$)

The $e$-quotient $Q_e(\lambda )=({\lambda }^{(0)},\dots ,{\lambda }^{(e-1)})$ is the tuple of partitions represented by the individual runners after their positions are divided by $e$.

Hooks of $\lambda$ whose lengths are divisible by $e$ correspond bijectively to hooks in the $e$-quotient. A bead-gap pair on one runner with distance $eh$ becomes a bead-gap pair of distance $h$ in that runner partition, and hook removal commutes with this correspondence.