A Bijection between Atomic Partitions and Unsplitable Partitions

TL;DR

This paper establishes a bijection between atomic partitions and unsplitable partitions within the algebra of noncommutative symmetric functions, connecting two different free generating sets.

Contribution

It provides the first explicit bijection between atomic and unsplitable partitions, linking two known bases of the algebra of noncommutative symmetric functions.

Findings

Constructed a bijection between atomic and unsplitable partitions.

Bridged two different free generating sets of NCSym.

Enhanced understanding of the combinatorial structure of NCSym.

Abstract

In the study of the algebra $\mathrm{NCSym}$ of symmetric functions in noncommutative variables, Bergeron and Zabrocki found a free generating set consisting of power sum symmetric functions indexed by atomic partitions. On the other hand, Bergeron, Reutenauer, Rosas, and Zabrocki studied another free generating set of $\mathrm{NCSym}$ consisting of monomial symmetric functions indexed by unsplitable partitions. Can and Sagan raised the question of finding a bijection between atomic partitions and unsplitable partitions. In this paper, we provide such a bijection.