Run Distribution Over Flattened Partitions

Abstract

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.