A bijection between atomic partitions and unsplitable partitions (Q625367)

Summary: In the study of the algebra NCSym of symmetric functions in noncommutative variables, \textit{N. Bergeron} and \textit{M. Zabrocki} [``The Hopf algebras of symmetric functions and quasi-symmetric functions in non-commutative variables are free and co-free,'' J. Algebra Appl. 8, No.\,4, 581--600 (2009; Zbl 1188.16030)] found a free generating set consisting of power sum symmetric functions indexed by atomic partitions. On the other hand, \textit{N. Bergeron}, \textit{C. Reutenauer}, \textit{M. Rosas}, and \textit{M. Zabrocki} [``Invariants and coinvariants of the symmetric group in noncommuting variables,'' Can. J. Math. 60, No.\,2, 266--296 (2008; Zbl 1180.16025)] studied another free generating set of NCSym consisting of monomial symmetric functions indexed by unsplitable partitions. \textit{M.B. Can} and \textit{B.E. Sagan} [``Partitions, rooks, and symmetric functions in noncommuting variables,'' Electron. J. Comb. 18, No.\,2, Research Paper P3, 7 p., electronic only (2011; Zbl 1223.05008)] raised the question of finding a bijection between atomic partitions and unsplitable partitions. In this paper, we provide such a bijection.