Schur positivity of nabla on Petrie symmetric functions

The Petrie symmetric function $G(k,n)$, introduced by Grinberg, is defined as the sum of monomial symmetric functions $m_\lambda$ indexed by partitions $\lambda\vdash n$ satisfying $\lambda_1<k$.

This article demonstrates that the Schur positivity pattern of $\nabla^r G(k,n)$ for all $r\geq 1$ depends exclusively on whether $k$ divides $n$, thus answering an open problem of Bergeron noted in Grinberg's work.