  For all positive integers $k,l,n$, the Little Glaisher theorem states that
the number of partitions of $n$ into parts not divisible by $k$ and occurring
less than $l$ times is equal to the number of partitions of $n$ into parts not
divisible by $l$ and occurring less than $k$ times. While this refinement of
Glaisher theorem is easy to establish by computation of the generating
function, there is still no one-to-one canonical correspondence explaining it.
Our paper brings an answer to this open problem through an arithmetical
approach. Furthermore, in the case $l=2$, we discuss the possibility of
constructing a Schur-type companion of the Little Glaisher theorem via the
weighted words.
