 Glaisher partition bijection

Fix a prime $p$. Replace each part $p^{a}m$, with $p\nmid m$, by $p^a$ copies of $m$. This is a size-preserving bijection from regular partitions to partitions of an integer whose parts are not divisible by $p$. To invert it, write each multiplicity of a part $m$ not divisible by $p$ in base $p$ and use its digit in position $a$ as the multiplicity of $p^{a}m$. Every resulting multiplicity is below $p$. The bijection explains why the number of simple symmetric-group modules equals the number of p-regular elements' conjugacy classes.