  The study of flattened partitions is an active area of current research. In
this paper, our study unexpectedly leads us to the OEIS numbers A124324. We
provide a new combinatorial interpretation of these numbers. A combinatorial
bijection between flattened partitions over $[n+1]$ and the partitions of $[n]$
is also given in a separate section. We introduce the numbers $f_{n, k}$ which
count the number of flattened partitions over $[n]$ having $k$ runs. We give
recurrence relations defining them, as well as their exponential generating
function in differential form. It should be appreciated if its closed form is
established. We extend the results to flattened partitions where the first $s$
integers belong to different runs. Combinatorial proofs are given.
