= Core-quotient bijection for partitions
Fix a positive integer $e$. A <partition of an integer> is uniquely specified by its <core of a partition> for modulus $e$ and its ordered <quotient of a partition> for that modulus. More precisely, for every $e$-core $\gamma$ and ordered tuple $(\nu_0,\ldots,\nu_{e-1})$ of partitions, there is a unique partition $\lambda$ such that
$$
C_e(\lambda)=\gamma,\qquad Q_e(\lambda)=(\nu_0,\ldots,\nu_{e-1}),
\qquad |\lambda|=|\gamma|+e\sum_i|\nu_i|.
$$
The ordered runners use the usual <abacus of a partition> convention with a number of beads divisible by $e$. Adding $e$ beads preserves both the tuple and its runner labels. The core fixes the relative runner offsets, and each component partition fixes the displacement of the beads on that runner, giving existence and uniqueness.
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