I am wondering if the following problem has already been approached.
Given a unit length $l$, a segment of length $L$, integer, is also given (call it the master segment), as well as $m$ segments of lengths $m_i$, $i$ running from $1$ to $m$ and all the $m_i$ integer too. The length of all m segments is less than the master segment
The latter segments can be placed on the segment long $N$ only in such a way that the unit length long interval overlap (in other words, each segment can start and finish only at integer values of length of the master segment).
How many ways are there to place all the $m$ segments?