Run Distribution Over Flattened Partitions

TL;DR

This paper explores flattened partitions, introduces new combinatorial interpretations and bijections, and studies the enumeration of such partitions with various constraints, connecting to OEIS sequence A124324.

Contribution

It provides a new combinatorial interpretation of OEIS sequence A124324 and introduces recurrence relations and generating functions for counting flattened partitions.

Findings

Connected flattened partitions to OEIS A124324

Derived recurrence relations for counting flattened partitions

Extended results to partitions with initial elements in different runs

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.